{"id":6766,"date":"2020-10-09T18:03:56","date_gmt":"2020-10-09T18:03:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6766"},"modified":"2026-02-01T07:21:00","modified_gmt":"2026-02-01T07:21:00","slug":"5-6-multiplying-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/5-6-multiplying-polynomials\/","title":{"raw":"5.6: Multiplying Polynomials","rendered":"5.6: Multiplying Polynomials"},"content":{"raw":"
\r\n

section 5.6 Learning Objectives<\/h3>\r\n5.6: Multiplying Polynomials<\/strong>\r\n
    \r\n \t
  • Find the product of monomials<\/li>\r\n \t
  • Find the product of a monomial and a polynomial<\/li>\r\n \t
  • Find the product of two binomials\r\n
      \r\n \t
    • Using the Distributive Property<\/li>\r\n \t
    • Using the FOIL Method<\/li>\r\n \t
    • Using the Table Method<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
    • Square a binomial<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

      Find the product of monomials<\/h2>\r\nMultiplying polynomials<\/strong> involves applying the rules of exponents and the distributive property to simplify the product. Polynomial\u00a0multiplication can be useful in modeling real world situations. Understanding polynomial products is an important step in learning to solve algebraic equations involving polynomials. There are many, varied uses for polynomials including the generation of 3D graphics for entertainment and industry, as in the image below.\r\n\r\n[caption id=\"attachment_4567\" align=\"aligncenter\" width=\"431\"]\"Bumpy, Surfaces made from polynomials with AutoCAD[\/caption]\r\n\r\nIn the exponents section, we\u00a0practiced\u00a0multiplying\u00a0monomials together, like we did with this expression: [latex]24{x}^{8}\\cdot2{x}^{5}[\/latex]. The only thing different between that section and this one is that we called it simplifying, and now we are calling it polynomial multiplication. \u00a0Remember that simplifying a mathematical expression means performing as many operations as we can until there are no more to perform, including multiplication. \u00a0In this section we will show examples of how to multiply more than just monomials. \u00a0We will multiply monomials with\u00a0binomials and trinomials. We will also learn some techniques for multiplying two binomials together.\r\n
      \r\n

      Example 1<\/h3>\r\nMultiply. [latex]-9x^{3}\\cdot 3x^{2}[\/latex]\r\n\r\n[reveal-answer q=\"322242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"322242\"]\r\n\r\nRearrange the factors.\r\n

      [latex]-9\\cdot3\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\r\nMultiply constants. Remember that a positive number times a negative number yields a negative number.\r\n

      [latex]-27\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\r\nMultiply variable terms. Remember to add the exponents when multiplying exponents with the same base.\r\n

      [latex]\\begin{array}{l}-27\\cdot x^{3+2}\\\\-27\\cdot x^{5}\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex]-9x^{3}\\cdot 3x^{2}=-27x^{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThat\u2019s it! When multiplying monomials, multiply the coefficients together, and then multiply the variables together. Remember, if two variables have the same base, follow the rules of exponents, like this:\r\n

      [latex] \\displaystyle 5{{a}^{4}}\\cdot 7{{a}^{6}}=35{{a}^{10}}[\/latex]<\/p>\r\nThe following video provides more examples of multiplying monomials with different exponents.\r\n\r\nhttps:\/\/youtu.be\/30x8hY32B0o\r\n

      Find the product of a monomial and a polynomial<\/h2>\r\nThe distributive property can be used to multiply a monomial and a binomial. Just remember that the monomial must be multiplied by each term in the binomial. In the next example, you will see how to multiply a second degree monomial with a binomial. \u00a0Note the use of exponent rules.\r\n
      \r\n

      Example 2<\/h3>\r\nSimplify. [latex]5x^2\\left(4x^{2}+3x\\right)[\/latex]\r\n[reveal-answer q=\"176215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"176215\"]Distribute the monomial to each term of the binomial. Multiply coefficients and variables separately.\r\n

      [latex]\\begin{array}{c}5x^2\\left(4x^{2}\\right)+5x^2\\left(3x\\right)\\\\\\text{ }\\\\=20x^{2+2}+15x^{2+1}\\\\\\text{ }\\\\=20x^{4}+15x^{3}\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex]5x^2\\left(4x^{2}+3x\\right)=20x^{4}+15x^{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow let's add another layer by multiplying a monomial by a trinomial. Consider the expression [latex]2x\\left(2x^{2}+5x+10\\right)[\/latex].\r\n\r\nThis expression can be modeled with a sketch like the one below.\r\n\r\n\"Area\r\n

      The only difference between this example and the previous one is there is one more term to distribute the monomial to.<\/p>\r\n

      [latex]\\begin{array}{c}2x\\left(2x^{2}+5x+10\\right)=2x\\left(2x^{2}\\right)+2x\\left(5x\\right)=2x\\left(10\\right)\\\\=4x^{3}+10x^{2}+20x\\end{array}[\/latex]<\/p>\r\nYou will always need to pay attention to negative signs when you are multiplying. Watch\u00a0what happens to the sign on the terms in the trinomial when it is multiplied by a negative monomial in the next example.\r\n

      \r\n

      Example 3<\/h3>\r\nSimplify. [latex]-7x\\left(2x^{2}-5x+1\\right)[\/latex]\r\n[reveal-answer q=\"590272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"590272\"]\r\n\r\nDistribute the monomial to each term in the trinomial.\r\n

      [latex]-7x\\left(2x^{2}\\right)-7x\\left(-5x\\right)-7x\\left(1\\right)[\/latex]<\/p>\r\nMultiply.\r\n

      [latex]\\begin{array}{c}-14x^{1+2}+35x^{1+1}-7x\\\\\\text{ }\\\\-14x^{3}+35x^{2}-7x\\end{array}[\/latex]<\/p>\r\nRewrite addition of terms with negative coefficients as subtraction.\r\n

      Answer<\/h4>\r\n[latex]-7x\\left(2x^{2}-5x+1\\right)=-14x^{3}+35x^{2}-7x[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      The following video provides more examples of multiplying a monomial and a polynomial.<\/p>\r\nhttps:\/\/youtu.be\/bwTmApTV_8o\r\n

      Find the product of two binomials<\/h2>\r\nNow let's explore multiplying two binomials. For those of you that use pictures to learn, you can draw an area model to help make sense of the process. You'll use each binomial as one of the dimensions of a rectangle, and their product as the area.\r\n\r\nThe model below shows [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]:\r\n\r\n\"Area\r\n\r\nEach binomial is expanded into variable terms and constants, [latex]x+4[\/latex], along the top of the model and [latex]x+2[\/latex] along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, [latex]x^{2}+2x+4x+8[\/latex], If you combine all the like terms, you can write the product, or area, as [latex]x^{2}+6x+8[\/latex].\r\n

      Find the product of two binomials using the Distributive Property<\/h3>\r\nYou can use the distributive property to determine the product of two binomials.\r\n
      \r\n

      Example 4<\/h3>\r\nSimplify. [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]\r\n[reveal-answer q=\"186797\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"186797\"]Distribute the [latex]x[\/latex] over [latex]x+2[\/latex], then distribute 4 over [latex]x+2[\/latex].\r\n

      [latex]x\\left(x\\right)+x\\left(2\\right)+4\\left(x\\right)+4\\left(2\\right)[\/latex]<\/p>\r\nMultiply.\r\n

      [latex]x^{2}+2x+4x+8[\/latex]<\/p>\r\nCombine like terms [latex]\\left(2x+4x\\right)[\/latex].\r\n

      [latex]x^{2}+6x+8[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex]\\left(x+4\\right)\\left(x+2\\right)=x^{2}+6x+8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nLook back at the model above to see where each piece of [latex]x^{2}+2x+4x+8[\/latex] comes from. Can you see where you multiply [latex]x[\/latex] by [latex]x + 2[\/latex], and where you get [latex]x^{2}[\/latex]\u00a0from [latex]x\\left(x\\right)[\/latex]?\r\n\r\nAnother way to look at multiplying binomials is to see that each term in one binomial is multiplied by each term in the other binomial. Look at the example above: the [latex]x[\/latex] in [latex]x+4[\/latex] gets multiplied by both the [latex]x[\/latex] and the 2 from [latex]x+2[\/latex], and the 4 gets multiplied by both the [latex]x[\/latex] and the 2.\r\n\r\nThe following video provides an example of multiplying two binomials using an area model as well as repeated distribution.\r\n\r\nhttps:\/\/youtu.be\/u4Hgl0BrUlo\r\n\r\nNext we will explore other methods for multiplying two binomials, and\u00a0become aware of the different forms that binomials can have.\r\n

      Find the product of two binomials using the FOIL Method<\/h3>\r\n[caption id=\"attachment_4589\" align=\"aligncenter\" width=\"335\"]\"Crane Foil Crane[\/caption]\r\n\r\nWe just looked at the example of multiplying two binomials, [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. In this section we will provide examples of how to use two different methods to multiply to binomials. Keep in mind as you read through the page that simplify and multiply are used interchangeably.\r\n\r\nSome people use the FOIL method to keep track of which pairs of terms have been multiplied when you are multiplying two binomials. This is not the same thing you use to wrap up leftovers, but an acronym for First, Outer, Inner, Last.<\/strong> Let's go back to a previous example, where we were asked to multiply the two binomials: [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. \u00a0The following steps show you how to apply this method to multiplying two binomials.\r\n

      [latex]\\begin{array}{l}\\text{First}\\text{ term in each binomial}: \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(2\\right)=2x\\\\\\text{Inner terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(x\\right)=4x\\\\\\text{Last terms in each binomial}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(2\\right)=8\\end{array}[\/latex]<\/p>\r\nWhen you add the four results, you get the same answer,\u00a0[latex]x^{2}+2x+4x+8=x^{2}+6x+8[\/latex].\r\nThe last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.\r\n

      \"Caution\"Caution! Note that the FOIL method only works for multiplying two binomials together. It does not work for multiplying a binomial and a trinomial, or two trinomials.<\/div>\r\n\r\n[caption id=\"attachment_4595\" align=\"alignleft\" width=\"138\"]\"two Order Doesn't Matter When You Multiply[\/caption]\r\n\r\n \r\n\r\nOne of the neat things about multiplication is that\u00a0terms can be multiplied in either order. The expression [latex]\\left(x+2\\right)\\left(x+4\\right)[\/latex] has the same product as [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. They are both equal to [latex]x^{2}+6x+8[\/latex]. (Work it out and see.) The order in which you multiply binomials does not matter. What matters is that you multiply each term in one binomial by each term in the other binomial.\r\n\r\n \r\n\r\n \r\n\r\nIn the next example, you will see that sometimes there are constants in front of the variable. They will get multiplied together just as we have done before.\r\n
      \r\n

      Example 5<\/h3>\r\nSimplify [latex]\\left(4x\u201310\\right)\\left(2x+3\\right)[\/latex] using the FOIL acronym.\r\n\r\n[reveal-answer q=\"930433\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"930433\"]\r\n

      [latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,4x\\left(2x\\right)=8x^{2}\\\\\\text{Outer}:\\,\\,\\,4x\\left(3\\right)=12x\\\\\\text{Inner}:\\,\\,\\,\u221210\\left(2x\\right)=-20x\\\\\\text{Last}:\\,\\,\\,\\,\\,-10\\left(3\\right)=-30\\end{array}[\/latex]<\/p>\r\nBe careful about including the negative sign on the [latex]\u201110[\/latex], since 10 is subtracted.\r\n\r\nCombine like terms.\r\n

      [latex]\\begin{array}{c}8x^{2}+12x\u201320x\u201330\\\\\\text{ }\\\\=8x^{2}-8x\u201330\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex]\\left(4x\u201310\\right)\\left(2x+3\\right)=8x^{2}\u20138x\u201330[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe video that follows gives another example of multiplying two binomials using the FOIL acronym. Remember this method only works when you are multiplying two binomials.\r\n\r\nhttps:\/\/youtu.be\/_MrdEFnXNGA\r\n\r\nPolynomials can take many forms. \u00a0So far we have seen examples of binomials with variable terms on the left and constant terms on the right, such as this binomial [latex]\\left(2r-3\\right)[\/latex]. \u00a0Variables may also be on the right of the constant term, as in this binomial [latex]\\left(5+r\\right)[\/latex]. \u00a0In the next example, we will show that multiplying binomials in this form requires one extra step at the end.\u00a0 We will also demonstrate how to use another method to multiply binomials.\u00a0 It is called the Table Method.\r\n

      Find the product of two binomials using the Table Method<\/h3>\r\n
      \r\n

      Example 6<\/h3>\r\nFind the product.[latex]\\left(3\u2013s\\right)\\left(1-s\\right)[\/latex]\r\n[reveal-answer q=\"531601\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"531601\"]\r\n\r\nNotice how the binomials have the variable on the right instead of the left. \u00a0There is nothing different in the way you find the product. \u00a0At the end we will reorganize terms so they are in descending order as a matter of convention.\r\n\r\n[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)[\/latex]\r\n\r\nUse a table this time.\r\n\r\n\r\n\r\n\r\n\r\n
      <\/td>\r\n[latex]3[\/latex]<\/th>\r\n[latex]-s[\/latex]<\/th>\r\n<\/tr>\r\n
      [latex]1[\/latex]<\/th>\r\n[latex]3[\/latex]<\/td>\r\n[latex]-s[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]-s[\/latex]<\/th>\r\n[latex]-3s[\/latex]<\/td>\r\n[latex]s^2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice how the s<\/em> term is now positive. Collect the terms and simplify.\r\n

      [latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=3-3s-s+s^2\\\\\\text{ }\\\\=3-4s+s^2\\end{array}[\/latex]<\/p>\r\nAs a matter of convention, we will organize the terms so the one with greatest degree comes first. Pay close attention to the signs on the terms when you reorganize them. The 3 is positive, so we will use a plus in front of it, and the 4 is negative so we use a minus in front of it.\r\n

      [latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=s^{2}-4s+3\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)=s^2-4s+3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next two examples, we want to show you another common form a binomial can take.\u00a0 Each of the terms in the two binomials is the same, but the signs are different. You will see that in this case, the middle term will disappear.\r\n
      \r\n

      Example 7<\/h3>\r\nMultiply the binomials. [latex]\\left(x+8\\right)\\left(x\u20138\\right)[\/latex]\r\n[reveal-answer q=\"812247\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"812247\"]\r\n

      [latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer}:\\,\\,\\,\\,\\,\\,x\\left(-8\\right)=-8x\\\\\\text{Inner}:\\,\\,\\,\\,\\,\\,\\,8\\left(x\\right)=+8x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\left(-8\\right)=-64\\end{array}[\/latex]<\/p>\r\nAdd the terms. Note how the two x terms are opposites, so their sum is zero.\r\n

      [latex]\\begin{array}{c}x^{2}\\underbrace{-8x+8x}-64\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\text{ }\\\\=x^2-64\\\\\\text{ }\\\\\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex]\\left(x+8\\right)\\left(x-8\\right)=x^{2}-64[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/tWsLJ_pn5mQ\r\n

      Square a binomial<\/h2>\r\nIn the next few examples, we will look at what happens when a binomial is squared.\r\n\r\nThe expression [latex]{\\left(x+3\\right)}^{2}[\/latex] means the same thing as [latex]\\left(x+3\\right)\\left(x+3\\right)[\/latex]. \u00a0To find this product, let's use the table method. We will place the terms of each binomial along the top row and first column of a table, like this:\r\n\r\n\r\n\r\n\r\n\r\n
      <\/td>\r\n[latex]x[\/latex]<\/td>\r\n[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]x[\/latex]<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      [latex]+3[\/latex]<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      <\/td>\r\n[latex]x[\/latex]<\/td>\r\n[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n<\/thead>\r\n
      [latex]x[\/latex]<\/td>\r\n[latex]x\\cdot{x}=x^2[\/latex]<\/span><\/td>\r\n[latex]3\\cdot{x}=+3x[\/latex]<\/span><\/td>\r\n<\/tr>\r\n
      [latex]+3[\/latex]<\/td>\r\n[latex]x\\cdot{3}=+3x[\/latex]<\/span><\/td>\r\n\u00a0[latex]3\\cdot{3}=+9[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow we can write the terms of the polynomial from the entries in the table:\r\n

      [latex]\\left(x+3\\right)^{2}[\/latex]<\/p>\r\n

      =\u00a0[latex]x^2[\/latex]<\/span> + [latex]3x[\/latex]<\/span> + [latex]3x[\/latex]<\/span> + [latex]9[\/latex]<\/span><\/p>\r\n

      = [latex]x^{2}[\/latex]<\/span> + [latex]6x[\/latex]<\/span> + [latex]9[\/latex]<\/span>.<\/p>\r\n\r\n

      \r\n

      Example 8<\/h3>\r\nFind the product.\u00a0[latex]\\left(2x+6\\right)^{2}[\/latex]\r\n[reveal-answer q=\"255359\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"255359\"]We will use the FOIL method.\r\n[latex]\\left(2x+6\\right)^{2}=\\left(2x+6\\right)\\left(2x+6\\right)[\/latex]\r\n[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,2x\\left(2x\\right)=4x^{2}\\\\\\text{Outer}:\\,\\,\\,2x\\left(6\\right)=12x\\\\\\text{Inner}:\\,\\,\\,6\\left(2x\\right)=12x\\\\\\text{Last}:\\,\\,\\,\\,\\,6\\left(6\\right)=36\\end{array}[\/latex]\r\n\r\nNow you can collect the terms and simplify:\r\n[latex]\\begin{array}{c}4x^2+12x+12x+36\\\\\\text{ }\\\\=4x^2+24x+36\\end{array}[\/latex]\r\n\r\nAnswer\r\n\r\n[latex](2x+6)^{2}=4x^{2}+24x+36[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
      \r\n

      Example 9<\/h3>\r\nSquare the binomial difference\u00a0[latex]\\left(x\u20137\\right)^{2}[\/latex]\r\n[reveal-answer q=\"293164\"]Show solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"293164\"]\r\n\r\nWrite the product of the binomial.\r\n

      [latex]{\\left(x-7\\right)}^2=\\left(x\u20137\\right)\\left(x\u20137\\right)[\/latex]<\/p>\r\n

      Let's use the table method, just because. Note how we carry the negative sign with the\u00a0[latex]7[\/latex].<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      <\/td>\r\n\u00a0[latex]x[\/latex]<\/td>\r\n\u00a0[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n
      \u00a0[latex]x[\/latex]<\/td>\r\n\u00a0\u00a0[latex]x^2[\/latex]<\/td>\r\n\u00a0\u00a0[latex]-7x[\/latex]<\/td>\r\n<\/tr>\r\n
      \u00a0\u00a0[latex]-7[\/latex]<\/td>\r\n\u00a0\u00a0[latex]-7x[\/latex]<\/td>\r\n\u00a0\u00a0[latex]49[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCollect the terms, and simplify. Note how we keep the sign on each term.\r\n

      [latex]\\begin{array}{c}x^2-7x-7x+49\\\\\\text{ }\\\\=x^2-14x+49\\end{array}[\/latex]<\/p>\r\nAnswer\r\n[latex]x^2-14x+49[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      \r\n\r\n\"Caution\"Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:\r\n

      You can't move the exponent into a grouped sum because of the order of operations!!!!!<\/p>\r\n

      INCORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}\\neq2^{2}+x^{2}[\/latex]<\/p>\r\n

      \u00a0CORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}=\\left(2+x\\right)\\left(2+x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\nThe following video works through two more examples of squaring a binomial.\r\n\r\nhttps:\/\/youtu.be\/l7ivdpf4XnQ\r\n

      \r\n

      Think About It<\/h3>\r\n

      There are predictable outcomes when you square a binomial sum or difference. In general terms, for a binomial difference,<\/p>\r\n

      [latex]\\left(a-b\\right)^{2}=\\left(a-b\\right)\\left(a-b\\right)[\/latex],<\/p>\r\n

      the resulting product, after being simplified, will look like this:<\/p>\r\n

      [latex]a^2-2ab+b^2[\/latex].<\/p>\r\n

      The product of a binomial sum will have the following predictable outcome:<\/p>\r\n

      [latex]\\left(a+b\\right)^{2}=\\left(a+b\\right)\\left(a+b\\right)=a^2+2ab+b^2[\/latex].<\/p>\r\n

      Note that a and b in these generalizations could be integers, fractions, or variables with any kind of constant. \u00a0You will learn more about predictable patterns from products of binomials in later math classes.<\/p>\r\n\r\n<\/div>\r\nWe have looked at two\u00a0methods for multiplying two binomials together, the FOIL method and the Table method. \u00a0Why are we focusing\u00a0so much on binomials? \u00a0They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them.\u00a0Some of the forms a product of two binomials can take are listed here:\r\n

        \r\n \t
      • [latex]\\left(x+5\\right)\\left(2x-3\\right)[\/latex]<\/li>\r\n \t
      • [latex]\\left(x+7\\right)^{2}[\/latex]<\/li>\r\n \t
      • [latex]\\left(x-1\\right)^{2}[\/latex]<\/li>\r\n \t
      • [latex]\\left(2-y\\right)\\left(5+y\\right)[\/latex]<\/li>\r\n \t
      • [latex]\\left(x+9\\right)\\left(x-9\\right)[\/latex]<\/li>\r\n \t
      • [latex]\\left(2x-4\\right)\\left(x+3\\right)[\/latex]<\/li>\r\n<\/ul>\r\nAnd this is just a small list, the possible combinations are endless. For each of the products in the list, using one of the two methods presented here will work to simplify.\r\n

        Summary<\/h2>\r\nMultiplication of binomials and polynomials requires an understanding of the distributive property, rules for exponents, and a keen eye for collecting like terms. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in simplified form if all of its like terms have been combined.\r\n

        <\/h2>","rendered":"
        \n

        section 5.6 Learning Objectives<\/h3>\n

        5.6: Multiplying Polynomials<\/strong><\/p>\n

          \n
        • Find the product of monomials<\/li>\n
        • Find the product of a monomial and a polynomial<\/li>\n
        • Find the product of two binomials\n
            \n
          • Using the Distributive Property<\/li>\n
          • Using the FOIL Method<\/li>\n
          • Using the Table Method<\/li>\n<\/ul>\n<\/li>\n
          • Square a binomial<\/li>\n<\/ul>\n<\/div>\n

             <\/p>\n

            Find the product of monomials<\/h2>\n

            Multiplying polynomials<\/strong> involves applying the rules of exponents and the distributive property to simplify the product. Polynomial\u00a0multiplication can be useful in modeling real world situations. Understanding polynomial products is an important step in learning to solve algebraic equations involving polynomials. There are many, varied uses for polynomials including the generation of 3D graphics for entertainment and industry, as in the image below.<\/p>\n

            \"Bumpy,<\/p>\n

            Surfaces made from polynomials with AutoCAD<\/p>\n<\/div>\n

            In the exponents section, we\u00a0practiced\u00a0multiplying\u00a0monomials together, like we did with this expression: [latex]24{x}^{8}\\cdot2{x}^{5}[\/latex]. The only thing different between that section and this one is that we called it simplifying, and now we are calling it polynomial multiplication. \u00a0Remember that simplifying a mathematical expression means performing as many operations as we can until there are no more to perform, including multiplication. \u00a0In this section we will show examples of how to multiply more than just monomials. \u00a0We will multiply monomials with\u00a0binomials and trinomials. We will also learn some techniques for multiplying two binomials together.<\/p>\n

            \n

            Example 1<\/h3>\n

            Multiply. [latex]-9x^{3}\\cdot 3x^{2}[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            Rearrange the factors.<\/p>\n

            [latex]-9\\cdot3\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\n

            Multiply constants. Remember that a positive number times a negative number yields a negative number.<\/p>\n

            [latex]-27\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\n

            Multiply variable terms. Remember to add the exponents when multiplying exponents with the same base.<\/p>\n

            [latex]\\begin{array}{l}-27\\cdot x^{3+2}\\\\-27\\cdot x^{5}\\end{array}[\/latex]<\/p>\n

            Answer<\/h4>\n

            [latex]-9x^{3}\\cdot 3x^{2}=-27x^{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

            That\u2019s it! When multiplying monomials, multiply the coefficients together, and then multiply the variables together. Remember, if two variables have the same base, follow the rules of exponents, like this:<\/p>\n

            [latex]\\displaystyle 5{{a}^{4}}\\cdot 7{{a}^{6}}=35{{a}^{10}}[\/latex]<\/p>\n

            The following video provides more examples of multiplying monomials with different exponents.<\/p>\n