{"id":1656,"date":"2022-04-19T21:48:21","date_gmt":"2022-04-19T21:48:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1656"},"modified":"2026-01-06T20:22:15","modified_gmt":"2026-01-06T20:22:15","slug":"3-6-algebraic-analysis-on-intersection-points","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/3-6-algebraic-analysis-on-intersection-points\/","title":{"raw":"3.6: Algebraic Analysis on Intersection Points","rendered":"3.6: Algebraic Analysis on Intersection Points"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-154\" class=\"standard post-154 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Explain the meaning of solving a polynomial equation graphically<\/li>\r\n \t<li>Solve polynomial equations using the zero product property<\/li>\r\n \t<li>Find the [latex]x[\/latex]-intercepts of a polynomial function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>The Meaning of Solving a Polynomial Equation<\/h2>\r\nSuppose we have two functions [latex]f(x)=x^2-2x-10[\/latex] and [latex]g(x)=5[\/latex]. Using Desmos to graph these functions we get the graph in figure 1.\r\n\r\n[caption id=\"attachment_1897\" align=\"aligncenter\" width=\"202\"]<img class=\"wp-image-1897\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29001954\/desmos-graph-87-300x300.png\" alt=\"Intersection of two functions. The horizontal line, and the parabola given above.\" width=\"202\" height=\"202\" \/> Figure 1. The intersection of two functions[\/caption]\r\n\r\nThe horizontal line representing [latex]g(x)=5[\/latex] intersects the parabola representing [latex]f(x)=x^2-2x-10[\/latex] in two points. The two graphs intersect at the points (\u20133, 5) and (5, 5). These are the points where [latex]f(x)=g(x)[\/latex] or, in this case, where [latex]x^2-2x-10=5[\/latex]. In other words, the [latex]x[\/latex]-coordinates of the intersection points are solutions of the equation [latex]x^2-2x-10=5[\/latex]. Finding the intersection points graphically, as in figure 1, is equivalent to solving the equation [latex]f(x)=g(x)[\/latex].\r\n\r\nThe intersection point(s) between the graphs of any two functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex] can be found algebraically by setting the two functions equal to each other:\r\n<p style=\"text-align: center;\">[latex]f(x) = g(x)[\/latex]<\/p>\r\nAt any intersection point [latex](x,y)[\/latex], the value of [latex]x[\/latex] is the same for both functions, as is the value of [latex]y[\/latex]. In other words, [latex]f(x)=g(x)[\/latex] means when the two functions have the same input [latex]x[\/latex], and the output of the two functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are equal (i.e., [latex]f(x)=g(x)[\/latex]).\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nUse <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos<\/a> to find the solution to the equation [latex]x^2-x-12=0[\/latex].\r\n<h4>Solution<\/h4>\r\nLet [latex]f(x)=x^2-x-12[\/latex] and [latex]g(x)=0[\/latex]. Note that\u00a0[latex]g(x)=0[\/latex] is represented by the [latex]x[\/latex]-axis, so we are finding where the graph of [latex]y=f(x)[\/latex] intersects the [latex]x[\/latex]-axis (i.e., the [latex]x[\/latex]-intercepts).\r\n\r\nUsing Desmos we get:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1898 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88-300x300.png\" alt=\"Intersection of a parabola with the x-axis, with the functions of each given above, and intersections given below.\" width=\"300\" height=\"300\" \/><\/p>\r\nFrom the graph, [latex]y=f(x)[\/latex] intersects the [latex]x[\/latex]-axis at (\u20133, 0) and (4, 0).\r\n\r\nSo the solution to the equation\u00a0[latex]x^2-x-12=0[\/latex] is [latex]x=\u20133,\\;4[\/latex].\r\n\r\n&nbsp;\r\n\r\n<strong>Note.<\/strong> We can always check our solution by substituting the values into the equation:\r\n\r\nWhen [latex]x=\u20133[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x^2-x-12&amp;=0\\\\(-3)^2-(-3)-12&amp;=0\\\\9+3-12&amp;=0\\\\0&amp;=0\\;\\;\\;\\;\\text{True}\\end{aligned}[\/latex]<\/p>\r\nWhen [latex]x=4[\/latex],\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x^2-x-12&amp;=0\\\\(4)^2-(4)-12&amp;=0\\\\16-4-12&amp;=0\\\\0&amp;=0\\;\\;\\;\\;\\text{True}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nUse<a href=\"https:\/\/www.desmos.com\/calculator\"> Desmos<\/a> to find the solution to the equation [latex]x^2-4=-x^2+14[\/latex].\r\n<h4>Solution<\/h4>\r\nLet [latex]f(x)=x^2-4[\/latex] and [latex]g(x)=-x^2+14[\/latex]. then use Desmos to graph the two functions.\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1899 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89-300x300.png\" alt=\"Intersection of the two functions given above, showing the points of intersection given below.\" width=\"300\" height=\"300\" \/><\/p>\r\nFrom the graph, the intersection points are (\u20133, 5) and (3, 5).\r\n\r\nSo, the solution of the equation\u00a0[latex]x^2-4=-x^2+14[\/latex] is [latex]x=\u20133,\\;3[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nUse <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos<\/a> to find the solution to the equation [latex]x^2-3=6[\/latex].\r\n\r\n[reveal-answer q=\"hjm655\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm655\"]\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1900\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29005842\/desmos-graph-90-300x300.png\" alt=\"Intersection of two functions f of x equals x squared minus 3, and g of x equals 6. The two functions have points of intersection of (-3,6) and (3,6)\" width=\"160\" height=\"160\" \/><\/p>\r\n[latex]x=-3,\\;3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>The Relationship of Equivalent Equations, Solutions, Zeros, [latex]x[\/latex]-Intercepts, and Points of Intersection<\/h3>\r\nNo matter the equation, [latex]f(x)=g(x)[\/latex] can always be simplified to [latex]f(x)-g(x)=0[\/latex]. The difference in two polynomial functions is always another polynomial function, so solving\u00a0[latex]f(x)=g(x)[\/latex] is equivalent to solving\u00a0[latex]p(x)=0[\/latex], where\u00a0[latex]p(x)=f(x)-g(x)[\/latex].\r\n\r\nFor example, the equation in Example 2,\u00a0[latex]x^2-4=-x^2+14[\/latex], can be simplified to [latex]2x^2-18=0[\/latex]. [latex]x^2-4=-x^2+14[\/latex] and [latex]2x^2-18=0[\/latex] are <em><strong>equivalent equations\u00a0<\/strong><\/em>so they have identical solutions. Consequently,\u00a0solving\u00a0[latex]x^2-4=-x^2+14[\/latex] is equivalent to solving [latex]2x^2-18=0[\/latex], which in turn is equivalent to finding the <em><strong>zeros of the function<\/strong><\/em> [latex]p(x)=2x^2-18[\/latex]. In other words, solving\u00a0[latex]x^2-4=-x^2+14[\/latex] is equivalent to finding the <em><strong>intersection<\/strong><\/em> of\u00a0[latex]p(x)=2x^2-18[\/latex] with the [latex]x[\/latex]-axis, which is where the <em><strong>[latex]x[\/latex]-intercepts<\/strong><\/em> lie. Figure 2 shows this graphically.\r\n\r\n[caption id=\"attachment_1906\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1906 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-91-300x300.png\" alt=\"A graphical example that solving the equation in example 2 is the same as finding the zeros of the equivalent function.\" width=\"300\" height=\"300\" \/> Figure 2. Finding zeros of a function[\/caption]\r\n\r\nThe zeros of\u00a0[latex]p(x)=2x^2-18[\/latex] are [latex]x=-3,\\;3[\/latex].\r\n\r\nThe [latex]x[\/latex]-intercepts of the graph of\u00a0[latex]p(x)=2x^2-18[\/latex] are (\u20133, 0) and (3, 0).\r\n\r\nThe graphs of\u00a0[latex]f(x)=2x^2-18[\/latex] and [latex]g(x)=0[\/latex] intersect at [latex]x=-3,\\;3[\/latex].\r\n\r\nThe solutions of the polynomial equation\u00a0[latex]2x^2-18=0[\/latex] are\u00a0[latex]x=-3,\\;3[\/latex].\r\n\r\nDo you see the relationship between zeros, intersection points, [latex]x[\/latex]-intercepts, and solutions?\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nFind the zeros of the polynomial function [latex]f(x)=x^3+2x^2-8x[\/latex].\r\n<h4>Solution<\/h4>\r\nFinding the zeros is equivalent to solving the equation\u00a0[latex]x^3+2x^2-8x=0[\/latex].\r\n\r\nGraphically, this is equivalent to finding where the function\u00a0[latex]f(x)=x^3+2x^2-8x[\/latex] intersects the [latex]x[\/latex]-axis (the line [latex]g(x)=0[\/latex]).\r\n<p style=\"text-align: left;\">Using Desmos, we graph\u00a0[latex]f(x)=x^3+2x^2-8x[\/latex] and [latex]g(x)=0[\/latex].<\/p>\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/1ytdigxfvr?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\r\nThe curve intersects the horizontal line at (\u20134, 0), (0, 0) and (2, 0).\r\n\r\nConsequently, the zeros of\u00a0[latex]f(x)=x^3+2x^2-8x[\/latex] are [latex]x=-4,\\;0,\\;2[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nUse <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos<\/a> to find the zeros of the polynomial function [latex]f(x)=x^3-2x^2-3x[\/latex].\r\n\r\n[reveal-answer q=\"hjm495\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm495\"]\r\n\r\n[latex]x=-1,\\;0,\\;3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Solving a Polynomial Equation<\/h2>\r\nWe now know what it means to solve a polynomial equation graphically, but what about solving the equation algebraically? Afterall, the intersection points of two curves will not always end up on integer values of [latex]x[\/latex].\r\n\r\nOne way to solve a polynomial set equal to zero is to use the <em><strong>zero product property<\/strong><\/em>.\r\n<h3>The Zero Product Property<\/h3>\r\nSuppose we multiply two numbers together and get an answer of zero. What can you say about the two numbers? Could they be\u00a0[latex]2[\/latex] and\u00a0[latex]5[\/latex]? Could they be\u00a0[latex]9[\/latex] and\u00a0[latex]\u20131[\/latex]? No! The only way to get a product of zero, is to multiply by zero. So if a product equals zero, at least one of the factors must be zero. This idea is called the <em><strong>zero product property<\/strong><\/em>, and it is useful for solving polynomial\u00a0equations that can be factored.\r\n<div class=\"textbox shaded\">\r\n<h3>Zero Product property<\/h3>\r\nThe Zero Product Property states that if the product of two or more factors is\u00a0[latex]0[\/latex], then at least one of the factors must be [latex]0[\/latex].\r\n\r\nIf [latex]ab=0[\/latex], then either [latex]a=0[\/latex] or [latex]b=0[\/latex], or both [latex]a[\/latex]\u00a0and [latex]b=0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nSolve the equation [latex]x(x-3)=0[\/latex].\r\n<h4>Solution<\/h4>\r\nAccording to the zero product property,\r\n\r\n[latex]x(x-3)=0[\/latex] means [latex]x=0[\/latex] or [latex]x-3=0[\/latex]\r\n\r\nIf [latex]x-3=0[\/latex], then [latex]x=3[\/latex].\r\n<h4>Answer<\/h4>\r\n[latex]x=0,\\;3[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nSolve the equation [latex](2x-5)(7x+6)(3x+4)=0[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(2x-5)(7x+6)(3x+4)&amp;=0\\\\2x-5=0\\;\\;\\text{ or }\\;\\;7x+6=0\\;\\;\\text{ or }\\;\\;3x+4&amp;=0\\\\2x=5\\;\\;\\;\\;\\;7x=-6\\;\\;\\;\\;\\;3x&amp;=-4\\\\x=\\dfrac{5}{2}\\;\\;\\;\\;\\;x=-\\dfrac{6}{7}\\;\\;\\;\\;\\;x&amp;=-\\dfrac{4}{3}\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nSolve the equation:\r\n<ol>\r\n \t<li>\u00a0[latex](x+1)(x-7)=0[\/latex]<\/li>\r\n \t<li>\u00a0[latex]x(3x-5)=0[\/latex]<\/li>\r\n \t<li>\u00a0[latex]-5x(3x+1)(2x-9)=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm837\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm837\"]\r\n\r\n1.\r\n\r\n[latex]x=-1,\\;7[\/latex]\r\n\r\n2.\r\n\r\n[latex]x=0,\\;\\dfrac{5}{3}[\/latex]\r\n\r\n3.\r\n\r\n[latex]x=0,\\;-\\dfrac{1}{3},\\;\\dfrac{9}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Using Factoring to Solve a Polynomial Equation<\/h3>\r\nThe zero product property makes it possible for us to solve factored polynomial equations. So, if the equation contains a polynomial that is not factored, the first thing we need to do is 1) simplify it so that it is equal to zero, and 2) factor.\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nSolve the equation\u00a0[latex]x^2-x-12=0[\/latex].\r\n<h4>Solution<\/h4>\r\nThe equation is already set equal to zero, so all we have to do is factor it, then use the zero product property to solve it:\r\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{aligned}x^2-x-12&amp;=0\\\\(x-4)(x+3)&amp;=0\\\\x-4=0\\;\\;\\text{or}\\;\\;x+3&amp;=0\\\\x=4\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;x&amp;=-3\\end{aligned}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]x=4,\\;-3[\/latex]\r\n\r\n&nbsp;\r\n\r\nGraphically, this means that the graph of the function [latex]f(x)=x^2-x-12[\/latex] has [latex]x[\/latex]-intercepts at (4, 0) and (\u20133, 0). See example 1.\r\n\r\n<\/div>\r\nThe following video shows two more examples of using both factoring and the principle of zero products to solve a polynomial equation.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/gIwMkTAclw8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/Transcript-3.6-1.odt\">Transcript 3.6-1<\/a>\r\n\r\nThe next video shows that we can use previously learned methods to factor a trinomial in order to solve a polynomial equation.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/bi7i_RuIGl0?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/Transcript-3.6-2.odt\">Transcript 3.6-2<\/a>\r\n\r\nWhat happens if we don't have zero on one side of the equation? More often than not, we will start by simplifying the equation to get zero on one side.\r\n<div class=\"textbox examples\">\r\n<h3>Example 6<\/h3>\r\nSolve: [latex]s^2-4s=5[\/latex]\r\n<h4>Solution<\/h4>\r\nTo solve a polynomial equation, we need to have a zero on one side of the equation so we can factor and use the zero product principle to solve the equation. So, we will start by subtracting 5 from both sides of the equation:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}s^2-4s&amp;=5\\\\s^2-4s-5&amp;=0\\\\\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We now have all the terms on the left side and zero on the right side. The polynomial [latex]s^2-4s-5[\/latex] factors nicely which makes this equation a good candidate for the zero product principle.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}s^2-4s-5&amp;=0\\\\\\left(s+1\\right)\\left(s-5\\right)&amp;=0\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We separate our factors into two linear equations using the zero product property.<\/p>\r\n[latex](s-5)=0[\/latex]\r\n\r\n[latex]s-5=0[\/latex]\r\n\r\n[latex]s=5[\/latex]\r\n\r\nOR\r\n\r\n[latex](s+1)=0[\/latex]\r\n\r\n[latex]s+1=0[\/latex]\r\n\r\n[latex]s=-1[\/latex]\r\n\r\nTherefore, [latex]s=-1\\text{ OR }s=5[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nSolve [latex]15x^2=16x+15[\/latex].\r\n\r\n[reveal-answer q=\"hjm067\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm067\"]\r\n\r\n[latex]x=-\\dfrac{3}{5},\\;\\dfrac{5}{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nLet's work through one more example that is similar to the one above, except this example has fractions, yay!\r\n<div class=\"textbox examples\">\r\n<h3>Example 7<\/h3>\r\nSolve [latex]y^2-5=-\\dfrac{7}{2}y+\\dfrac{5}{2}[\/latex].\r\n<h4>Solution<\/h4>\r\nWe can solve this equation by first multiplying the equation by a common denominator to get rid of the fractions.\u00a0Start by multiplying the whole equation by\u00a0[latex]2[\/latex] to eliminate the fractions:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}2\\left(y^2-5\\right)&amp;=2\\left(-\\dfrac{7}{2}y+\\dfrac{5}{2}\\right)\\\\2y^2-10&amp;=-7y+5\\end{aligned}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Now we can move all the terms to one side and see if the resulting polynomial will factor so we can use the zero product property:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}2y^2-10&amp;=-7y+5\\\\2y^2-10+7y-5&amp;=0\\\\2y^2-15+7y&amp;=0\\\\2y^2+7y-15&amp;=0\\end{aligned}[\/latex]<\/p>\r\nWe can now check whether this polynomial will factor. Using a table we can list factors until we find two numbers with a product of [latex]ac=2\\cdot(-15)=-30[\/latex] and a sum of [latex]b=7[\/latex].\r\n<table style=\"width: 20%;\" summary=\"A table with 7 rows and 2 columns. The first column is labeled: Factors of -30 while the second is labeled: Sum of Factors. The entries in the first column are: 1, -30; -1, 30; 2, -15; -2, 15; 3, -10; and -3, 10. The entries in the second column are: -29, 29, -13, 13, -7, and 7.\">\r\n<thead>\r\n<tr>\r\n<th>Factors of [latex]2\\cdot-15=-30[\/latex]<\/th>\r\n<th>Sum of Factors<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]1,-30[\/latex]<\/td>\r\n<td>[latex]-29[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1,30[\/latex]<\/td>\r\n<td>[latex]29[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2,-15[\/latex]<\/td>\r\n<td>[latex]-13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2,15[\/latex]<\/td>\r\n<td>[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3,-10[\/latex]<\/td>\r\n<td>[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><span style=\"color: #0000ff;\">[latex]-3,10[\/latex]<\/span><\/td>\r\n<td><span style=\"color: #0000ff;\">[latex]7[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[latex]10,-3[\/latex] multiply to \u201330 and add to 7. We replace [latex]7y[\/latex] with [latex]10y-3y[\/latex] then factor by grouping:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}2y^2+7y-15&amp;=0\\\\ 2y^2+10y-3y-15&amp;=0\\\\2y(y+5)-3(y+5)&amp;=0\\\\(y+5)(2y-3)&amp;=0\\end{aligned}[\/latex]<\/p>\r\nNow we can set each factor equal to zero and solve:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(2y-3)=0\\;\\;\\;\\text{ OR }\\;\\;\\;\\left(y+5\\right)&amp;=0\\\\2y=3\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;y&amp;=-5\\\\y=\\frac{3}{2}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;y&amp;=-5\\end{aligned}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nRemember, we can always check to make sure our solutions are correct:\r\n\r\nCheck [latex]y=\\frac{3}{2}[\/latex]\r\n\r\n[latex]\\begin{array}{ccc}\\left(\\frac{3}{2}\\right)^2-5=-\\frac{7}{2}\\left(\\frac{3}{2}\\right)+\\frac{5}{2}\\\\\\frac{9}{4}-5=-\\frac{21}{4}+\\frac{5}{2}\\\\\\text{ common denominator = 4}\\\\\\frac{9}{4}-\\frac{20}{4}=-\\frac{21}{4}+\\frac{10}{4}\\\\-\\frac{11}{4}=-\\frac{11}{4}\\end{array}[\/latex]\r\n\r\n[latex]y=\\frac{3}{2}[\/latex] is indeed a solution, now check\u00a0[latex]y=-5[\/latex]\r\n\r\n[latex]\\begin{array}{ccc}\\left(-5\\right)^2-5=-\\frac{7}{2}\\left(-5\\right)+\\frac{5}{2}\\\\25-5=\\frac{35}{2}+\\frac{5}{2}\\\\20=\\frac{40}{2}\\\\20=20\\end{array}[\/latex]\r\n\r\n[latex]y=-5[\/latex] is also a solution, so we must have done something right!\r\n<p style=\"text-align: left;\">Therefore, [latex]y=\\frac{3}{2}\\text{ OR }y=-5[\/latex].<\/p>\r\n\r\n<\/div>\r\nThe next video shows how to solve another quadratic equation that contains fractions.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/kDj_qdKW-ls?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/Transcript-3.6-3.odt\">Transcript 3.6-3<\/a>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nSolve [latex]\\dfrac{1}{2}x^2+\\dfrac{7}{6}x=1[\/latex]\r\n\r\n[reveal-answer q=\"hjm809\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm809\"]\r\n\r\n[latex]x=-3,\\;\\dfrac{2}{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThere are, of course, polynomial equations that do not factor; they are\u00a0<em><strong>prime<\/strong><\/em><em>. <\/em>But just because they are prime, does not mean that they do not intersect with the [latex]x[\/latex]-axis. For example, the function [latex]f(x)=x^2-3x-1[\/latex] is prime, yet it crosses the [latex]x[\/latex]-axis at two points. This means that the equation [latex]x^2-3x-1=0[\/latex] has two solutions (figure 3). We just can't find them by factoring.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">x-intercepts<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">No x-intercepts<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[caption id=\"attachment_1912\" align=\"aligncenter\" width=\"208\"]<img class=\"wp-image-1912 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29191721\/desmos-graph-94-300x300.png\" alt=\"Parabola rossing the x-axis in two points\" width=\"208\" height=\"208\" \/> Figure 3. A prime polynomial function with two [latex]x[\/latex]-intercepts[\/caption]<\/td>\r\n<td style=\"width: 50%;\">[caption id=\"attachment_1913\" align=\"aligncenter\" width=\"208\"]<img class=\"wp-image-1913 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29192200\/desmos-graph-95-300x300.png\" alt=\"Parabola that never crosses the x-axis\" width=\"208\" height=\"208\" \/> Figure 4. A prime polynomial function with no [latex]x[\/latex]-intercepts[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn addition, there are polynomial equations that have no solution. For example the function [latex]g(x)=x^2+2[\/latex] never crosses the [latex]x[\/latex]-axis, therefore there is no real solution of the equation\u00a0[latex]x^2+2=0[\/latex] (figure 4).\r\n\r\nThe techniques we have learned can be used to solve polynomial equations that factor, find zeros of functions, find intersection points of functions, and find [latex]x-intercepts[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 8<\/h3>\r\nFind the [latex]x[\/latex]-intercepts of the graph of the function [latex]f(x)=6x^2-11x-10[\/latex].\r\n<h4>Solution<\/h4>\r\nWe could do this graphically using Desmos, but we end up with fractional [latex]x[\/latex]-values. So we will work the problem algebraically.\r\n\r\n<img class=\"aligncenter wp-image-1908\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29180124\/desmos-graph-93-300x300.png\" alt=\"x-intercepts are unknown fractional values.\" width=\"175\" height=\"175\" \/>\r\n\r\n[latex]x[\/latex]-intercepts are found when [latex]f(x)=0[\/latex] so we need to solve the equation [latex]6x^2-11x-10 = 0[\/latex]\r\n\r\nTo solve the equation for [latex]x[\/latex] we can use factoring and the zero product property:\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}6x^2-11x-10 &amp;= 0\\\\(2x-5)(3x+2)&amp;=0\\\\2x-5=0\\;\\;\\;\\;\\;\\text{or}\\;\\;\\;\\;\\;3x+2&amp;=0\\\\x=\\dfrac{5}{2}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;x&amp;=-\\dfrac{2}{3}\\end{aligned}[\/latex]<\/p>\r\nTherefore, the [latex]x[\/latex]-intercepts of the function [latex]f(x)=x^2-3x-10[\/latex] are [latex]\\left(\\dfrac{5}{2}, 0\\right)[\/latex] and [latex]\\left(\u2013\\dfrac{2}{3}, 0\\right)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 9<\/h3>\r\nDetermine the zeros of the function [latex]f(x)=4x^3-36x[\/latex].\r\n<h4>Solution<\/h4>\r\nWe need to solve the equation\u00a0[latex]4x^3-36x=0[\/latex].\r\n\r\nWe start by factoring then use the zero product property.\r\n\r\nThere is a common factor of [latex]4x[\/latex]:\u00a0\u00a0[latex]4x^3-36x=4x(x^2-9)[\/latex].\r\n\r\n[latex]x^2-9[\/latex] is the difference of two squares [latex]x^2-3^2=(x-3)(x+3)[\/latex].\r\n\r\nSo,\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}4x^3-36x&amp;=0\\\\4x(x-3)(x+3)&amp;=0\\\\4x=0\\;\\;\\;\\text{or}\\;\\;\\;x-3=0\\;\\;\\;\\text{or}\\;\\;\\;x+3&amp;=0\\\\x=0\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;x=3\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\,x&amp;=-3\\end{aligned}[\/latex]<\/p>\r\nThe zeros of\u00a0[latex]f(x)=4x^3-36x[\/latex] are [latex]x=0,\\;3,\\;-3[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nFind the [latex]x[\/latex]-intercepts of the graph of the function [latex]f(x)=15x^3+28x^2-32x[\/latex]\r\n\r\n[reveal-answer q=\"hjm111\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm111\"]\r\n\r\n[latex]\\left(-\\dfrac{8}{3},0\\right)[\/latex], [latex]\\left(0,0\\right)[\/latex] and\u00a0[latex]\\left(\\dfrac{4}{5},0\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 7<\/h3>\r\nSolve [latex](x-2)^2=(x-2)(x+5)[\/latex].\r\n\r\n[reveal-answer q=\"hjm276\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm276\"]\r\n\r\n[latex]x=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-154\" class=\"standard post-154 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Explain the meaning of solving a polynomial equation graphically<\/li>\n<li>Solve polynomial equations using the zero product property<\/li>\n<li>Find the [latex]x[\/latex]-intercepts of a polynomial function<\/li>\n<\/ul>\n<\/div>\n<h2>The Meaning of Solving a Polynomial Equation<\/h2>\n<p>Suppose we have two functions [latex]f(x)=x^2-2x-10[\/latex] and [latex]g(x)=5[\/latex]. Using Desmos to graph these functions we get the graph in figure 1.<\/p>\n<div id=\"attachment_1897\" style=\"width: 212px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1897\" class=\"wp-image-1897\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29001954\/desmos-graph-87-300x300.png\" alt=\"Intersection of two functions. The horizontal line, and the parabola given above.\" width=\"202\" height=\"202\" \/><\/p>\n<p id=\"caption-attachment-1897\" class=\"wp-caption-text\">Figure 1. The intersection of two functions<\/p>\n<\/div>\n<p>The horizontal line representing [latex]g(x)=5[\/latex] intersects the parabola representing [latex]f(x)=x^2-2x-10[\/latex] in two points. The two graphs intersect at the points (\u20133, 5) and (5, 5). These are the points where [latex]f(x)=g(x)[\/latex] or, in this case, where [latex]x^2-2x-10=5[\/latex]. In other words, the [latex]x[\/latex]-coordinates of the intersection points are solutions of the equation [latex]x^2-2x-10=5[\/latex]. Finding the intersection points graphically, as in figure 1, is equivalent to solving the equation [latex]f(x)=g(x)[\/latex].<\/p>\n<p>The intersection point(s) between the graphs of any two functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex] can be found algebraically by setting the two functions equal to each other:<\/p>\n<p style=\"text-align: center;\">[latex]f(x) = g(x)[\/latex]<\/p>\n<p>At any intersection point [latex](x,y)[\/latex], the value of [latex]x[\/latex] is the same for both functions, as is the value of [latex]y[\/latex]. In other words, [latex]f(x)=g(x)[\/latex] means when the two functions have the same input [latex]x[\/latex], and the output of the two functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are equal (i.e., [latex]f(x)=g(x)[\/latex]).<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Use <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos<\/a> to find the solution to the equation [latex]x^2-x-12=0[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Let [latex]f(x)=x^2-x-12[\/latex] and [latex]g(x)=0[\/latex]. Note that\u00a0[latex]g(x)=0[\/latex] is represented by the [latex]x[\/latex]-axis, so we are finding where the graph of [latex]y=f(x)[\/latex] intersects the [latex]x[\/latex]-axis (i.e., the [latex]x[\/latex]-intercepts).<\/p>\n<p>Using Desmos we get:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1898 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88-300x300.png\" alt=\"Intersection of a parabola with the x-axis, with the functions of each given above, and intersections given below.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-88.png 2000w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>From the graph, [latex]y=f(x)[\/latex] intersects the [latex]x[\/latex]-axis at (\u20133, 0) and (4, 0).<\/p>\n<p>So the solution to the equation\u00a0[latex]x^2-x-12=0[\/latex] is [latex]x=\u20133,\\;4[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Note.<\/strong> We can always check our solution by substituting the values into the equation:<\/p>\n<p>When [latex]x=\u20133[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x^2-x-12&=0\\\\(-3)^2-(-3)-12&=0\\\\9+3-12&=0\\\\0&=0\\;\\;\\;\\;\\text{True}\\end{aligned}[\/latex]<\/p>\n<p>When [latex]x=4[\/latex],<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x^2-x-12&=0\\\\(4)^2-(4)-12&=0\\\\16-4-12&=0\\\\0&=0\\;\\;\\;\\;\\text{True}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Use<a href=\"https:\/\/www.desmos.com\/calculator\"> Desmos<\/a> to find the solution to the equation [latex]x^2-4=-x^2+14[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Let [latex]f(x)=x^2-4[\/latex] and [latex]g(x)=-x^2+14[\/latex]. then use Desmos to graph the two functions.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1899 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89-300x300.png\" alt=\"Intersection of the two functions given above, showing the points of intersection given below.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89-1024x1024.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-89.png 2000w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>From the graph, the intersection points are (\u20133, 5) and (3, 5).<\/p>\n<p>So, the solution of the equation\u00a0[latex]x^2-4=-x^2+14[\/latex] is [latex]x=\u20133,\\;3[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Use <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos<\/a> to find the solution to the equation [latex]x^2-3=6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm655\">Show Answer<\/span><\/p>\n<div id=\"qhjm655\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1900\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29005842\/desmos-graph-90-300x300.png\" alt=\"Intersection of two functions f of x equals x squared minus 3, and g of x equals 6. The two functions have points of intersection of (-3,6) and (3,6)\" width=\"160\" height=\"160\" \/><\/p>\n<p>[latex]x=-3,\\;3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>The Relationship of Equivalent Equations, Solutions, Zeros, [latex]x[\/latex]-Intercepts, and Points of Intersection<\/h3>\n<p>No matter the equation, [latex]f(x)=g(x)[\/latex] can always be simplified to [latex]f(x)-g(x)=0[\/latex]. The difference in two polynomial functions is always another polynomial function, so solving\u00a0[latex]f(x)=g(x)[\/latex] is equivalent to solving\u00a0[latex]p(x)=0[\/latex], where\u00a0[latex]p(x)=f(x)-g(x)[\/latex].<\/p>\n<p>For example, the equation in Example 2,\u00a0[latex]x^2-4=-x^2+14[\/latex], can be simplified to [latex]2x^2-18=0[\/latex]. [latex]x^2-4=-x^2+14[\/latex] and [latex]2x^2-18=0[\/latex] are <em><strong>equivalent equations\u00a0<\/strong><\/em>so they have identical solutions. Consequently,\u00a0solving\u00a0[latex]x^2-4=-x^2+14[\/latex] is equivalent to solving [latex]2x^2-18=0[\/latex], which in turn is equivalent to finding the <em><strong>zeros of the function<\/strong><\/em> [latex]p(x)=2x^2-18[\/latex]. In other words, solving\u00a0[latex]x^2-4=-x^2+14[\/latex] is equivalent to finding the <em><strong>intersection<\/strong><\/em> of\u00a0[latex]p(x)=2x^2-18[\/latex] with the [latex]x[\/latex]-axis, which is where the <em><strong>[latex]x[\/latex]-intercepts<\/strong><\/em> lie. Figure 2 shows this graphically.<\/p>\n<div id=\"attachment_1906\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1906\" class=\"wp-image-1906 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-91-300x300.png\" alt=\"A graphical example that solving the equation in example 2 is the same as finding the zeros of the equivalent function.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-91-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-91-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-91-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-91-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-91-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-91-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/desmos-graph-91.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1906\" class=\"wp-caption-text\">Figure 2. Finding zeros of a function<\/p>\n<\/div>\n<p>The zeros of\u00a0[latex]p(x)=2x^2-18[\/latex] are [latex]x=-3,\\;3[\/latex].<\/p>\n<p>The [latex]x[\/latex]-intercepts of the graph of\u00a0[latex]p(x)=2x^2-18[\/latex] are (\u20133, 0) and (3, 0).<\/p>\n<p>The graphs of\u00a0[latex]f(x)=2x^2-18[\/latex] and [latex]g(x)=0[\/latex] intersect at [latex]x=-3,\\;3[\/latex].<\/p>\n<p>The solutions of the polynomial equation\u00a0[latex]2x^2-18=0[\/latex] are\u00a0[latex]x=-3,\\;3[\/latex].<\/p>\n<p>Do you see the relationship between zeros, intersection points, [latex]x[\/latex]-intercepts, and solutions?<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Find the zeros of the polynomial function [latex]f(x)=x^3+2x^2-8x[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Finding the zeros is equivalent to solving the equation\u00a0[latex]x^3+2x^2-8x=0[\/latex].<\/p>\n<p>Graphically, this is equivalent to finding where the function\u00a0[latex]f(x)=x^3+2x^2-8x[\/latex] intersects the [latex]x[\/latex]-axis (the line [latex]g(x)=0[\/latex]).<\/p>\n<p style=\"text-align: left;\">Using Desmos, we graph\u00a0[latex]f(x)=x^3+2x^2-8x[\/latex] and [latex]g(x)=0[\/latex].<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/1ytdigxfvr?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\n<p>The curve intersects the horizontal line at (\u20134, 0), (0, 0) and (2, 0).<\/p>\n<p>Consequently, the zeros of\u00a0[latex]f(x)=x^3+2x^2-8x[\/latex] are [latex]x=-4,\\;0,\\;2[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Use <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos<\/a> to find the zeros of the polynomial function [latex]f(x)=x^3-2x^2-3x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm495\">Show Answer<\/span><\/p>\n<div id=\"qhjm495\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=-1,\\;0,\\;3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Solving a Polynomial Equation<\/h2>\n<p>We now know what it means to solve a polynomial equation graphically, but what about solving the equation algebraically? Afterall, the intersection points of two curves will not always end up on integer values of [latex]x[\/latex].<\/p>\n<p>One way to solve a polynomial set equal to zero is to use the <em><strong>zero product property<\/strong><\/em>.<\/p>\n<h3>The Zero Product Property<\/h3>\n<p>Suppose we multiply two numbers together and get an answer of zero. What can you say about the two numbers? Could they be\u00a0[latex]2[\/latex] and\u00a0[latex]5[\/latex]? Could they be\u00a0[latex]9[\/latex] and\u00a0[latex]\u20131[\/latex]? No! The only way to get a product of zero, is to multiply by zero. So if a product equals zero, at least one of the factors must be zero. This idea is called the <em><strong>zero product property<\/strong><\/em>, and it is useful for solving polynomial\u00a0equations that can be factored.<\/p>\n<div class=\"textbox shaded\">\n<h3>Zero Product property<\/h3>\n<p>The Zero Product Property states that if the product of two or more factors is\u00a0[latex]0[\/latex], then at least one of the factors must be [latex]0[\/latex].<\/p>\n<p>If [latex]ab=0[\/latex], then either [latex]a=0[\/latex] or [latex]b=0[\/latex], or both [latex]a[\/latex]\u00a0and [latex]b=0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Solve the equation [latex]x(x-3)=0[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>According to the zero product property,<\/p>\n<p>[latex]x(x-3)=0[\/latex] means [latex]x=0[\/latex] or [latex]x-3=0[\/latex]<\/p>\n<p>If [latex]x-3=0[\/latex], then [latex]x=3[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x=0,\\;3[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Solve the equation [latex](2x-5)(7x+6)(3x+4)=0[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(2x-5)(7x+6)(3x+4)&=0\\\\2x-5=0\\;\\;\\text{ or }\\;\\;7x+6=0\\;\\;\\text{ or }\\;\\;3x+4&=0\\\\2x=5\\;\\;\\;\\;\\;7x=-6\\;\\;\\;\\;\\;3x&=-4\\\\x=\\dfrac{5}{2}\\;\\;\\;\\;\\;x=-\\dfrac{6}{7}\\;\\;\\;\\;\\;x&=-\\dfrac{4}{3}\\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Solve the equation:<\/p>\n<ol>\n<li>\u00a0[latex](x+1)(x-7)=0[\/latex]<\/li>\n<li>\u00a0[latex]x(3x-5)=0[\/latex]<\/li>\n<li>\u00a0[latex]-5x(3x+1)(2x-9)=0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm837\">Show Answer<\/span><\/p>\n<div id=\"qhjm837\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p>[latex]x=-1,\\;7[\/latex]<\/p>\n<p>2.<\/p>\n<p>[latex]x=0,\\;\\dfrac{5}{3}[\/latex]<\/p>\n<p>3.<\/p>\n<p>[latex]x=0,\\;-\\dfrac{1}{3},\\;\\dfrac{9}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Using Factoring to Solve a Polynomial Equation<\/h3>\n<p>The zero product property makes it possible for us to solve factored polynomial equations. So, if the equation contains a polynomial that is not factored, the first thing we need to do is 1) simplify it so that it is equal to zero, and 2) factor.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>Solve the equation\u00a0[latex]x^2-x-12=0[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The equation is already set equal to zero, so all we have to do is factor it, then use the zero product property to solve it:<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{aligned}x^2-x-12&=0\\\\(x-4)(x+3)&=0\\\\x-4=0\\;\\;\\text{or}\\;\\;x+3&=0\\\\x=4\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;x&=-3\\end{aligned}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x=4,\\;-3[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Graphically, this means that the graph of the function [latex]f(x)=x^2-x-12[\/latex] has [latex]x[\/latex]-intercepts at (4, 0) and (\u20133, 0). See example 1.<\/p>\n<\/div>\n<p>The following video shows two more examples of using both factoring and the principle of zero products to solve a polynomial equation.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/gIwMkTAclw8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/Transcript-3.6-1.odt\">Transcript 3.6-1<\/a><\/p>\n<p>The next video shows that we can use previously learned methods to factor a trinomial in order to solve a polynomial equation.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/bi7i_RuIGl0?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/Transcript-3.6-2.odt\">Transcript 3.6-2<\/a><\/p>\n<p>What happens if we don&#8217;t have zero on one side of the equation? More often than not, we will start by simplifying the equation to get zero on one side.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 6<\/h3>\n<p>Solve: [latex]s^2-4s=5[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>To solve a polynomial equation, we need to have a zero on one side of the equation so we can factor and use the zero product principle to solve the equation. So, we will start by subtracting 5 from both sides of the equation:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}s^2-4s&=5\\\\s^2-4s-5&=0\\\\\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: left;\">We now have all the terms on the left side and zero on the right side. The polynomial [latex]s^2-4s-5[\/latex] factors nicely which makes this equation a good candidate for the zero product principle.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}s^2-4s-5&=0\\\\\\left(s+1\\right)\\left(s-5\\right)&=0\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: left;\">We separate our factors into two linear equations using the zero product property.<\/p>\n<p>[latex](s-5)=0[\/latex]<\/p>\n<p>[latex]s-5=0[\/latex]<\/p>\n<p>[latex]s=5[\/latex]<\/p>\n<p>OR<\/p>\n<p>[latex](s+1)=0[\/latex]<\/p>\n<p>[latex]s+1=0[\/latex]<\/p>\n<p>[latex]s=-1[\/latex]<\/p>\n<p>Therefore, [latex]s=-1\\text{ OR }s=5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Solve [latex]15x^2=16x+15[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm067\">Show Answer<\/span><\/p>\n<div id=\"qhjm067\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=-\\dfrac{3}{5},\\;\\dfrac{5}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Let&#8217;s work through one more example that is similar to the one above, except this example has fractions, yay!<\/p>\n<div class=\"textbox examples\">\n<h3>Example 7<\/h3>\n<p>Solve [latex]y^2-5=-\\dfrac{7}{2}y+\\dfrac{5}{2}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>We can solve this equation by first multiplying the equation by a common denominator to get rid of the fractions.\u00a0Start by multiplying the whole equation by\u00a0[latex]2[\/latex] to eliminate the fractions:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}2\\left(y^2-5\\right)&=2\\left(-\\dfrac{7}{2}y+\\dfrac{5}{2}\\right)\\\\2y^2-10&=-7y+5\\end{aligned}[\/latex]<\/p>\n<p style=\"text-align: left;\">Now we can move all the terms to one side and see if the resulting polynomial will factor so we can use the zero product property:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}2y^2-10&=-7y+5\\\\2y^2-10+7y-5&=0\\\\2y^2-15+7y&=0\\\\2y^2+7y-15&=0\\end{aligned}[\/latex]<\/p>\n<p>We can now check whether this polynomial will factor. Using a table we can list factors until we find two numbers with a product of [latex]ac=2\\cdot(-15)=-30[\/latex] and a sum of [latex]b=7[\/latex].<\/p>\n<table style=\"width: 20%;\" summary=\"A table with 7 rows and 2 columns. The first column is labeled: Factors of -30 while the second is labeled: Sum of Factors. The entries in the first column are: 1, -30; -1, 30; 2, -15; -2, 15; 3, -10; and -3, 10. The entries in the second column are: -29, 29, -13, 13, -7, and 7.\">\n<thead>\n<tr>\n<th>Factors of [latex]2\\cdot-15=-30[\/latex]<\/th>\n<th>Sum of Factors<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]1,-30[\/latex]<\/td>\n<td>[latex]-29[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1,30[\/latex]<\/td>\n<td>[latex]29[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2,-15[\/latex]<\/td>\n<td>[latex]-13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2,15[\/latex]<\/td>\n<td>[latex]13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3,-10[\/latex]<\/td>\n<td>[latex]-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #0000ff;\">[latex]-3,10[\/latex]<\/span><\/td>\n<td><span style=\"color: #0000ff;\">[latex]7[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>[latex]10,-3[\/latex] multiply to \u201330 and add to 7. We replace [latex]7y[\/latex] with [latex]10y-3y[\/latex] then factor by grouping:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}2y^2+7y-15&=0\\\\ 2y^2+10y-3y-15&=0\\\\2y(y+5)-3(y+5)&=0\\\\(y+5)(2y-3)&=0\\end{aligned}[\/latex]<\/p>\n<p>Now we can set each factor equal to zero and solve:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}(2y-3)=0\\;\\;\\;\\text{ OR }\\;\\;\\;\\left(y+5\\right)&=0\\\\2y=3\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;y&=-5\\\\y=\\frac{3}{2}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;y&=-5\\end{aligned}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Remember, we can always check to make sure our solutions are correct:<\/p>\n<p>Check [latex]y=\\frac{3}{2}[\/latex]<\/p>\n<p>[latex]\\begin{array}{ccc}\\left(\\frac{3}{2}\\right)^2-5=-\\frac{7}{2}\\left(\\frac{3}{2}\\right)+\\frac{5}{2}\\\\\\frac{9}{4}-5=-\\frac{21}{4}+\\frac{5}{2}\\\\\\text{ common denominator = 4}\\\\\\frac{9}{4}-\\frac{20}{4}=-\\frac{21}{4}+\\frac{10}{4}\\\\-\\frac{11}{4}=-\\frac{11}{4}\\end{array}[\/latex]<\/p>\n<p>[latex]y=\\frac{3}{2}[\/latex] is indeed a solution, now check\u00a0[latex]y=-5[\/latex]<\/p>\n<p>[latex]\\begin{array}{ccc}\\left(-5\\right)^2-5=-\\frac{7}{2}\\left(-5\\right)+\\frac{5}{2}\\\\25-5=\\frac{35}{2}+\\frac{5}{2}\\\\20=\\frac{40}{2}\\\\20=20\\end{array}[\/latex]<\/p>\n<p>[latex]y=-5[\/latex] is also a solution, so we must have done something right!<\/p>\n<p style=\"text-align: left;\">Therefore, [latex]y=\\frac{3}{2}\\text{ OR }y=-5[\/latex].<\/p>\n<\/div>\n<p>The next video shows how to solve another quadratic equation that contains fractions.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/kDj_qdKW-ls?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/04\/Transcript-3.6-3.odt\">Transcript 3.6-3<\/a><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Solve [latex]\\dfrac{1}{2}x^2+\\dfrac{7}{6}x=1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm809\">Show Answer<\/span><\/p>\n<div id=\"qhjm809\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=-3,\\;\\dfrac{2}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>There are, of course, polynomial equations that do not factor; they are\u00a0<em><strong>prime<\/strong><\/em><em>. <\/em>But just because they are prime, does not mean that they do not intersect with the [latex]x[\/latex]-axis. For example, the function [latex]f(x)=x^2-3x-1[\/latex] is prime, yet it crosses the [latex]x[\/latex]-axis at two points. This means that the equation [latex]x^2-3x-1=0[\/latex] has two solutions (figure 3). We just can&#8217;t find them by factoring.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">x-intercepts<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">No x-intercepts<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1912\" style=\"width: 218px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1912\" class=\"wp-image-1912\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29191721\/desmos-graph-94-300x300.png\" alt=\"Parabola rossing the x-axis in two points\" width=\"208\" height=\"208\" \/><\/p>\n<p id=\"caption-attachment-1912\" class=\"wp-caption-text\">Figure 3. A prime polynomial function with two [latex]x[\/latex]-intercepts<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1913\" style=\"width: 218px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1913\" class=\"wp-image-1913\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29192200\/desmos-graph-95-300x300.png\" alt=\"Parabola that never crosses the x-axis\" width=\"208\" height=\"208\" \/><\/p>\n<p id=\"caption-attachment-1913\" class=\"wp-caption-text\">Figure 4. A prime polynomial function with no [latex]x[\/latex]-intercepts<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In addition, there are polynomial equations that have no solution. For example the function [latex]g(x)=x^2+2[\/latex] never crosses the [latex]x[\/latex]-axis, therefore there is no real solution of the equation\u00a0[latex]x^2+2=0[\/latex] (figure 4).<\/p>\n<p>The techniques we have learned can be used to solve polynomial equations that factor, find zeros of functions, find intersection points of functions, and find [latex]x-intercepts[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 8<\/h3>\n<p>Find the [latex]x[\/latex]-intercepts of the graph of the function [latex]f(x)=6x^2-11x-10[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>We could do this graphically using Desmos, but we end up with fractional [latex]x[\/latex]-values. So we will work the problem algebraically.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1908\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/04\/29180124\/desmos-graph-93-300x300.png\" alt=\"x-intercepts are unknown fractional values.\" width=\"175\" height=\"175\" \/><\/p>\n<p>[latex]x[\/latex]-intercepts are found when [latex]f(x)=0[\/latex] so we need to solve the equation [latex]6x^2-11x-10 = 0[\/latex]<\/p>\n<p>To solve the equation for [latex]x[\/latex] we can use factoring and the zero product property:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}6x^2-11x-10 &= 0\\\\(2x-5)(3x+2)&=0\\\\2x-5=0\\;\\;\\;\\;\\;\\text{or}\\;\\;\\;\\;\\;3x+2&=0\\\\x=\\dfrac{5}{2}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;x&=-\\dfrac{2}{3}\\end{aligned}[\/latex]<\/p>\n<p>Therefore, the [latex]x[\/latex]-intercepts of the function [latex]f(x)=x^2-3x-10[\/latex] are [latex]\\left(\\dfrac{5}{2}, 0\\right)[\/latex] and [latex]\\left(\u2013\\dfrac{2}{3}, 0\\right)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 9<\/h3>\n<p>Determine the zeros of the function [latex]f(x)=4x^3-36x[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>We need to solve the equation\u00a0[latex]4x^3-36x=0[\/latex].<\/p>\n<p>We start by factoring then use the zero product property.<\/p>\n<p>There is a common factor of [latex]4x[\/latex]:\u00a0\u00a0[latex]4x^3-36x=4x(x^2-9)[\/latex].<\/p>\n<p>[latex]x^2-9[\/latex] is the difference of two squares [latex]x^2-3^2=(x-3)(x+3)[\/latex].<\/p>\n<p>So,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}4x^3-36x&=0\\\\4x(x-3)(x+3)&=0\\\\4x=0\\;\\;\\;\\text{or}\\;\\;\\;x-3=0\\;\\;\\;\\text{or}\\;\\;\\;x+3&=0\\\\x=0\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;x=3\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\,x&=-3\\end{aligned}[\/latex]<\/p>\n<p>The zeros of\u00a0[latex]f(x)=4x^3-36x[\/latex] are [latex]x=0,\\;3,\\;-3[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Find the [latex]x[\/latex]-intercepts of the graph of the function [latex]f(x)=15x^3+28x^2-32x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm111\">Show Answer<\/span><\/p>\n<div id=\"qhjm111\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\dfrac{8}{3},0\\right)[\/latex], [latex]\\left(0,0\\right)[\/latex] and\u00a0[latex]\\left(\\dfrac{4}{5},0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 7<\/h3>\n<p>Solve [latex](x-2)^2=(x-2)(x+5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm276\">Show Answer<\/span><\/p>\n<div id=\"qhjm276\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1656\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>3.6: Algebraic Analysis on Intersection Points. <strong>Authored by<\/strong>: Leo Chang and Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Examples 1, 2, 3, 4, 5, 8, 9; Figures 1, 2, 3, 4; Try Its: hjm276; hjm111; hjm801; hjm067; hjm837; hjm495; hjm655. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using Desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/desmos.com\">http:\/\/desmos.com<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Factor and Solve Quadratic Equation - Greatest Common Factor Only. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/gIwMkTAclw8\">https:\/\/youtu.be\/gIwMkTAclw8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Factor and Solve Quadratic Equations When A equals 1. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Provided by<\/strong>: https:\/\/youtu.be\/bi7i_RuIGl0. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solve a Quadratic Equations with Fractions by Factoring (a not 1). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/kDj_qdKW-ls\">https:\/\/youtu.be\/kDj_qdKW-ls<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revised and adapted: Unit 12: Factoring, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"3.6: Algebraic Analysis on Intersection Points\",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Factor and Solve Quadratic Equation - Greatest Common Factor Only\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/gIwMkTAclw8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Factor and Solve Quadratic Equations When A equals 1\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"https:\/\/youtu.be\/bi7i_RuIGl0\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Solve a Quadratic Equations with Fractions by Factoring (a not 1)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/kDj_qdKW-ls\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Revised and adapted: Unit 12: Factoring, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Examples 1, 2, 3, 4, 5, 8, 9; Figures 1, 2, 3, 4; Try Its: hjm276; hjm111; hjm801; hjm067; hjm837; hjm495; hjm655\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using Desmos graphing calculator\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"desmos.com\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1656","chapter","type-chapter","status-publish","hentry"],"part":1176,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1656","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":38,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1656\/revisions"}],"predecessor-version":[{"id":4795,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1656\/revisions\/4795"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/1176"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/1656\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=1656"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1656"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=1656"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=1656"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}