{"id":4427,"date":"2016-05-27T22:27:54","date_gmt":"2016-05-27T22:27:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/nrocarithmetic\/?post_type=chapter&p=4427"},"modified":"2023-07-25T04:16:50","modified_gmt":"2023-07-25T04:16:50","slug":"read-define-and-evaluate-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/read-define-and-evaluate-polynomials\/","title":{"raw":"5.4: Intro to Polynomials","rendered":"5.4: Intro to Polynomials"},"content":{"raw":"
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section 5.4 Learning Objectives<\/h3>\r\n5.4: Introduction to Polynomials<\/strong>\r\n
    \r\n \t
  • Determine if an expression is a polynomial<\/li>\r\n \t
  • Identify the characteristics of a polynomial<\/li>\r\n \t
  • Determine if a polynomial is a monomial, binomial, or trinomial<\/li>\r\n \t
  • Evaluate a polynomial for a specified value<\/li>\r\n \t
  • Simplify polynomials by combining like terms<\/li>\r\n \t
  • Determine the domain of a polynomial function<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\nPolynomials are algebraic expressions that are created by combining numbers and variables using arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation. You can create a polynomial<\/b> by adding or subtracting terms. Polynomials are very useful in applications from science and engineering to business. You may see a resemblance between expressions, which we have been studying in this course, and polynomials. \u00a0Polynomials are a special sub-group of mathematical expressions and equations.\r\n

    Determine if an expression is a polynomial<\/h2>\r\nThe following table is intended to help you tell the difference between what is a polynomial and what is not.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    IS a Polynomial<\/strong><\/td>\r\nIs NOT a Polynomial<\/strong><\/td>\r\nBecause<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
    [latex]2x^2-\\frac{1}{2}x -9[\/latex]<\/td>\r\n[latex]\\frac{2}{x^{2}}+x[\/latex]<\/td>\r\nPolynomials only have variables in the numerator<\/td>\r\n<\/tr>\r\n
    [latex]\\frac{y}{4}-y^3[\/latex]<\/td>\r\n[latex]\\frac{2}{y}+4[\/latex]<\/td>\r\nPolynomials only have variables in the numerator<\/td>\r\n<\/tr>\r\n
    [latex]\\sqrt{12}\\left(a\\right)+9[\/latex]<\/td>\r\n\u00a0[latex]\\sqrt{a}+7[\/latex]<\/td>\r\n\u00a0Variables under a root are not allowed in polynomials<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

    Identify the characteristics of a polynomial<\/h2>\r\nThe basic building block of a polynomial is a monomial<\/b>. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the coefficient<\/b>.\r\n\r\nExamples of monomials:\r\n
      \r\n \t
    • number: [latex]{2}[\/latex]<\/li>\r\n \t
    • variable: [latex]{x}[\/latex]<\/li>\r\n \t
    • product of number and variable: [latex]{2x}[\/latex]<\/li>\r\n \t
    • product of number and variable with an exponent: [latex]{2x}^{3}[\/latex]<\/li>\r\n<\/ul>\r\n\"The\r\n\r\nThe coefficient can be any real number, including 0. The exponent of the variable must be a whole number\u20140, 1, 2, 3, and so on. A monomial cannot have a variable in the denominator or a negative exponent.\r\n\r\nFor a monomial in one variable, the value of the exponent is called the degree<\/b> of the monomial. Based on our exponent rules in Section 5.1, recall that [latex]x^{0}=1[\/latex].\u00a0 So, a monomial with no variable actually has a degree of 0.\u00a0 For example, we could rewrite the monomial 3 as [latex]3x^{0}[\/latex].\r\n
      \r\n

      Example 1<\/h3>\r\nIdentify the coefficient, variable, and degree\u00a0of the variable for the following monomial terms:\r\n1) 9\r\n2) [latex]x[\/latex]\r\n3) [latex] \\displaystyle \\frac{3}{5}{{k}^{8}}[\/latex]\r\n\r\n[reveal-answer q=\"150661\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"150661\"]\r\n1)\r\n9 is a constant so there is no coefficient or variable. Since there is no variable, we consider the degree to be 0.\r\n\r\n2)\r\nThe variable is [latex]x[\/latex].\r\n\r\nThe exponent of [latex]x[\/latex] is 1 since [latex]x=x^{1}[\/latex].\u00a0So, the degree is 1.\r\n\r\nThe coefficient of [latex]x[\/latex] is 1 since [latex]x=1x^{1}[\/latex].\r\n\r\n3)\r\nThe variable is [latex]k[\/latex] .\r\n\r\nThe exponent of [latex]k[\/latex] is 8, so the degree is 8.\r\n\r\nThe coefficient of [latex]k^{8}[\/latex]\u00a0is [latex] \\displaystyle \\frac{3}{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nA polynomial is a monomial or the sum or difference of two or more monomials. Each monomial is called a term <\/b>of the polynomial.\r\n\r\nThe word \u201cpolynomial\u201d has the prefix, \u201cpoly,\u201d which means many. However, the word polynomial can be used for all numbers of terms, including only one term.\r\n\r\nBecause the exponent of the variable must be a whole number, monomials and polynomials cannot have a variable in the denominator.\r\n\r\nPolynomials can be classified by the degree of the polynomial. The degree of a polynomial<\/strong> is the degree of its highest-degree term. The coefficient of the highest degree term is called the leading coefficient<\/strong>.\u00a0\u00a0So the degree of [latex]2x^{3}+3x^{2}+8x+5[\/latex] is 3 and the leading coefficient is 2.\r\n\r\nA polynomial is said to be written in standard form (or \"descending order\") when the terms are arranged from the highest-degree to the lowest degree. When it is written in standard form it is easy to determine the degree of the polynomial.\r\n\r\nIf a polynomial contains a term with no variable, it is called the\u00a0constant term<\/strong>. If the polynomial is in standard form, the constant term appears at the end. If no constant term is explicitly given, then the constant term is 0.\r\n
      \r\n

      terminology<\/h3>\r\n
        \r\n \t
      • Polynomial\u00a0<\/strong>- a sum of monomials, each called a term of the polynomial<\/li>\r\n \t
      • Degree of a Polynomial<\/strong> - the degree of its highest-degree term<\/li>\r\n \t
      • Leading Coefficient\u00a0<\/strong>- the coefficient of the highest-degree term<\/li>\r\n \t
      • Constant Term\u00a0<\/strong>- the term with no variable (if no such term is written, it is 0)<\/li>\r\n<\/ul>\r\n<\/div>\r\n
        \r\n

        Example 2<\/h3>\r\nFor each polynomial, determine the number of terms, the degree of the polynomial, the leading coefficient, and the constant term.\r\n
          \r\n \t
        1. [latex]\\hspace{.05in} 7x^3-4x^2+5x+8[\/latex]<\/li>\r\n \t
        2. [latex]\\hspace{.05in}-3x^9+2x[\/latex]<\/li>\r\n \t
        3. [latex]\\hspace{.05in}x^{4}-x-1[\/latex]<\/li>\r\n \t
        4. [latex]\\hspace{.05in}13-x^2+2x+\\dfrac{3}{4}x^{5}+x^3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"238986\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"238986\"]\r\n\r\n1.\u00a0 [latex]\\hspace{.05in} 7x^3-4x^2+5x+8[\/latex]\r\n\r\nThe polynomial has 4 terms (the 4 terms being [latex]7x^3[\/latex], [latex]-4x^2[\/latex], [latex]5x[\/latex], and [latex]8[\/latex]).\r\n\r\nSince the largest degree is [latex]3[\/latex], the degree of the polynomial is [latex]3[\/latex].\r\n\r\nThe coefficient of the third degree term is [latex]7[\/latex], so the leading coefficient is [latex]7[\/latex].\r\n\r\nThe constant term is [latex]8[\/latex], as this is the term that does not contain a variable.\r\n\r\n2.\u00a0 [latex]\\hspace{.05in}-3x^9+2x[\/latex]\r\n\r\nThe polynomial has 2 terms.\r\n\r\nThe degree of the polynomial is [latex]9[\/latex].\r\n\r\nThe leading coefficient is [latex]-3[\/latex].\r\n\r\nSince no constant term is included, the constant term is [latex]0[\/latex].\r\n\r\n3.\u00a0 [latex]\\hspace{.05in}x^4-x-1[\/latex]\r\n\r\nThe polynomial has 3 terms.\r\n\r\nThe degree of the polynomial is [latex]4[\/latex].\r\n\r\nSince[latex]x^4=1x^4[\/latex], the leading coefficient is [latex]1[\/latex].\r\n\r\nThe constant term is [latex]-1[\/latex] (be sure to include the negative sign with the term).\r\n\r\n4.\u00a0 [latex]\\hspace{.05in}13-x^2+2x+\\dfrac{3}{4}x^5+x^3[\/latex]\r\n\r\nNote that the polynomial is not in standard form. While we could still identify the relevant features, you might find it easier to first rewrite the polynomial in descending order as\r\n

          [latex]\\dfrac{3}{4}x^5+x^3-x^2+2x+13[\/latex].<\/p>\r\nThe polynomial has 5 terms.\r\n\r\nThe degree of the polynomial is [latex]5[\/latex].\r\n\r\nThe leading coefficient is [latex]\\dfrac{3}{4}[\/latex].\r\n\r\nThe constant term is [latex]13[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

          Determine if a polynomial is a monomial, binomial, or trinomial<\/h2>\r\nSome polynomials have specific names indicated by their prefix.\r\n
            \r\n \t
          • Monomial<\/b>\u2014is a polynomial with exactly one term (\u201cmono\u201d\u2014means one)<\/li>\r\n \t
          • Binomial<\/b>\u2014is a polynomial with exactly two terms (\u201cbi\u201d\u2014means two)<\/li>\r\n \t
          • Trinomial<\/b>\u2014is a polynomial with exactly three terms (\u201ctri\u201d\u2014means three)<\/li>\r\n<\/ul>\r\nThe table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
            Monomials<\/b><\/td>\r\nBinomials<\/b><\/td>\r\nTrinomials<\/b><\/td>\r\nOther Polynomials<\/b><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
            15<\/td>\r\n[latex]3y+13[\/latex]<\/td>\r\n[latex]x^{3}-x^{2}+1[\/latex]<\/td>\r\n[latex]5x^{4}+3x^{3}-6x^{2}+2x[\/latex]<\/td>\r\n<\/tr>\r\n
            [latex] \\displaystyle \\frac{1}{2}x[\/latex]<\/td>\r\n[latex]4p-7[\/latex]<\/td>\r\n[latex]3x^{2}+2x-9[\/latex]<\/td>\r\n[latex]\\frac{1}{3}x^{5}-2x^{4}+\\frac{2}{9}x^{3}-x^{2}+4x-\\frac{5}{6}[\/latex]<\/td>\r\n<\/tr>\r\n
            [latex]-4y^{3}[\/latex]<\/td>\r\n[latex]3x^{2}+\\frac{5}{8}x[\/latex]<\/td>\r\n[latex]3y^{3}+y^{2}-2[\/latex]<\/td>\r\n[latex]3t^{3}-3t^{2}-3t-3[\/latex]<\/td>\r\n<\/tr>\r\n
            [latex]16n^{4}[\/latex]<\/td>\r\n[latex]14y^{3}+3y[\/latex]<\/td>\r\n[latex]a^{7}+2a^{5}-3a^{3}[\/latex]<\/td>\r\n[latex]q^{7}+2q^{5}-3q^{3}+q[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen the coefficient of a polynomial term is 0, you usually do not write the term at all (because 0 times anything is 0, and adding 0 doesn\u2019t change the value). The last binomial above could be written as a trinomial, [latex]14y^{3}+0y^{2}+3y[\/latex].\r\n
            \r\n

            Example 3<\/h3>\r\nFor the following expressions, determine whether they are a polynomial. If so, categorize them as a monomial, binomial, or trinomial.\r\n
              \r\n \t
            1. [latex]\\frac{x-3}{1-x}+x^2[\/latex]<\/li>\r\n \t
            2. [latex]t^2+2t-3[\/latex]<\/li>\r\n \t
            3. [latex]x^3+\\frac{x}{8}[\/latex]<\/li>\r\n \t
            4. [latex]\\frac{\\sqrt{y}}{2}-y-1[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"239104\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"239104\"]\r\n
                \r\n \t
              1. [latex]\\frac{x-3}{1-x}+x^2[\/latex] is not a polynomial because it violates the rule that polynomials cannot have variables in the denominator of a fraction.<\/li>\r\n \t
              2. [latex]t^2+2t-3[\/latex] is a polynomial because it is an expression whose terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. \u00a0There are three terms in this polynomial so it is a trinomial.<\/li>\r\n \t
              3. [latex]x^3+\\frac{x}{8}[\/latex]is a polynomial because it is an expression whose terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. \u00a0There are two terms in this polynomial so it is a binomial.<\/li>\r\n \t
              4. [latex]\\frac{\\sqrt{y}}{2}-y-1[\/latex]\u00a0is not a polynomial because it violates the rule that polynomials cannot have variables\u00a0under a root.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will be shown more examples of how to identify and categorize polynomials.\r\n\r\nhttps:\/\/youtu.be\/nPAqfuoSbPI\r\n

                Evaluate a polynomial for a specified value<\/h2>\r\nYou can evaluate polynomials just as you have been evaluating expressions and functions all along. (Recall, we evaluated functions back in Module 3). To evaluate an expression for a value of the variable, you substitute the value for the variable every time<\/i> it appears. Then use the order of operations to find the resulting value for the expression.\r\n
                \r\n

                Example 4<\/h3>\r\nGiven [latex]f(x)=3x^{2}-2x+1[\/latex], evaluate [latex]f(-1)[\/latex].\r\n\r\n[reveal-answer q=\"280466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"280466\"]Substitute [latex]-1[\/latex] for each x<\/i> in the polynomial.\r\n

                [latex]f(-1) = 3\\left(-1\\right)^{2}-2\\left(-1\\right)+1[\/latex]<\/p>\r\nFollowing the order of operations, evaluate exponents first.\r\n

                [latex]f(-1) =3\\left(1\\right)-2\\left(-1\\right)+1[\/latex]<\/p>\r\nMultiply 3 times 1, and then multiply [latex]-2[\/latex] times [latex]-1[\/latex].\r\n

                [latex]f(-1) =3+\\left(-2\\right)\\left(-1\\right)+1[\/latex]<\/p>\r\nChange the subtraction to addition of the opposite.\r\n

                [latex]f(-1) =3+2+1[\/latex]<\/p>\r\nFind the sum.\r\n

                Answer<\/h4>\r\n[latex]f(-1)=6[\/latex]. This answer could also be written as the ordered pair,\u00a0 [latex](-1, 6)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
                \r\n

                Example 5<\/h3>\r\nGiven [latex] f(p)= \\displaystyle -\\frac{2}{3}p^{4}+2p^{3}-p[\/latex], evaluate [latex]f(3)[\/latex].\r\n\r\n[reveal-answer q=\"745542\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"745542\"]Substitute 3 for each p<\/i> in the polynomial.\r\n

                [latex] f(3)= \\displaystyle -\\frac{2}{3}\\left(3\\right)^{4}+2\\left(3\\right)^{3}-3[\/latex]<\/p>\r\nFollowing the order of operations, evaluate exponents first and then multiply.\r\n

                [latex]f(3)= \\displaystyle -\\frac{2}{3}\\left(81\\right)+2\\left(27\\right)-3[\/latex]<\/p>\r\nAdd and then subtract to get [latex]-3[\/latex].\r\n

                [latex]f(3)=-54 + 54 \u2013 3[\/latex]<\/p>\r\n\r\n

                Answer<\/h4>\r\n[latex] f(3)=-3[\/latex]. This answer could also be written as the ordered pair,\u00a0[latex](3, -3)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video presents more examples of evaluating a polynomial for a given value.\r\n\r\nhttps:\/\/youtu.be\/2EeFrgQP1hM\r\n

                Simplify polynomials by combining like terms<\/h2>\r\n[caption id=\"attachment_4439\" align=\"aligncenter\" width=\"488\"]\"Apple Apple and Orange[\/caption]\r\n\r\nA polynomial may need to be simplified. One way to simplify a polynomial is to combine the like terms<\/b> if there are any. Two or more terms in a polynomial are like terms if they have the same variable (or variables) with the same exponent. For example, [latex]3x^{2}[\/latex] and [latex]-5x^{2}[\/latex] are like terms: They both have x<\/i> as the variable, and the exponent is 2 for each. However, [latex]3x^{2}[\/latex]\u00a0and [latex]3x[\/latex]\u00a0are not like terms, because their exponents are different.\r\n\r\nHere are some examples of terms that are alike and some that are unlike.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                Term<\/strong><\/td>\r\nLike Terms<\/strong><\/td>\r\nUNLike Terms<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
                [latex]a[\/latex]<\/td>\r\n[latex]3a, \\,\\,\\,-2a,\\,\\,\\, \\frac{1}{2}a[\/latex]<\/td>\r\n[latex]a^2,\\,\\,\\,\\frac{1}{a},\\,\\,\\, \\sqrt{a}[\/latex]<\/td>\r\n<\/tr>\r\n
                [latex]a^2[\/latex]<\/td>\r\n[latex]-5a^2,\\,\\,\\,\\frac{1}{4}a^2,\\,\\,\\, 0.56a^2[\/latex]<\/td>\r\n[latex]\\frac{1}{a^2},\\,\\,\\,\\sqrt{a^2},\\,\\,\\, a^3[\/latex]<\/td>\r\n<\/tr>\r\n
                [latex]ab[\/latex]<\/td>\r\n[latex]7ab,\\,\\,\\,0.23ab,\\,\\,\\,\\frac{2}{3}ab,\\,\\,\\,-ab[\/latex]<\/td>\r\n[latex]a^2b,\\,\\,\\,\\frac{1}{ab},\\,\\,\\,\\sqrt{ab} [\/latex]<\/td>\r\n<\/tr>\r\n
                [latex]ab^2[\/latex]<\/td>\r\n\u00a0[latex]4ab^2,\\,\\,\\, \\frac{ab^2}{7},\\,\\,\\,0.4ab^2,\\,\\,\\, -ab^2[\/latex]<\/td>\r\n\u00a0[latex]a^2b,\\,\\,\\, ab,\\,\\,\\,\\sqrt{ab^2},\\,\\,\\,\\frac{1}{ab^2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
                \r\n

                Example 6<\/h3>\r\nWhich of these terms are like terms?\r\n

                [latex]7x^{3}+7x+7y-8x^{3}+9y-3x^{2}+8y^{2}[\/latex]<\/p>\r\n[reveal-answer q=\"413363\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"413363\"]Like terms must have the same variables, so first identify which terms use the same variables.\r\n

                [latex]\\begin{array}{l}x:7x^{3}+7x-8x^{3}-3x^{2}\\\\y:7y+9y+8y^{2}\\end{array}[\/latex]<\/p>\r\nLike terms must also have the same exponents. Identify which terms with the same variables also use the same exponents.\r\n\r\nThe x<\/em> terms [latex]7x^{3}[\/latex]\u00a0and [latex]-8x^{3}[\/latex]\u00a0have the same exponent.\r\n\r\nThe y<\/em> terms [latex]7y[\/latex] and [latex]9y[\/latex] have the same exponent.\r\n

                Answer<\/h4>\r\n[latex]7x^{3}[\/latex] and [latex]-8x^{3}[\/latex] are like terms.\r\n\r\n[latex]7y[\/latex] and [latex]9y[\/latex]\u00a0are like terms.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can use the distributive property to simplify the sum of like terms. Recall that the distributive property states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products.\r\n

                [latex]2\\left(3+6\\right)=2\\left(3\\right)+2\\left(6\\right)[\/latex]<\/p>\r\nBoth expressions equal 18. So you can write the expression in whichever form is the most useful.\r\n\r\nLet\u2019s see how we can use this property to combine like terms.\r\n

                \r\n

                Example 7<\/h3>\r\nSimplify [latex]3x^{2}-5x^{2}[\/latex].\r\n\r\n[reveal-answer q=\"969840\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"969840\"][latex]3x^{2}[\/latex] and [latex]5x^{2}[\/latex]\u00a0<\/sup>are like terms.\r\n

                [latex]3\\left(x^{2}\\right)-5\\left(x^{2}\\right)[\/latex]<\/p>\r\nWe can rewrite the expression as the product of the difference.\r\n

                [latex]\\left(3-5\\right)\\left(x^{2}\\right)[\/latex]<\/p>\r\nCalculate [latex]3\u20135[\/latex].\r\n

                [latex]\\left(-2\\right)\\left(x^{2}\\right)[\/latex]<\/p>\r\nWrite the difference of [latex]3 \u2013 5[\/latex] as the new coefficient.\r\n

                Answer<\/h4>\r\n[latex]3x^{2}-5x^{2}=-2x^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may have noticed that combining like terms involves combining the coefficients to find the new coefficient of the like term. You can use this as a shortcut.\r\n
                \r\n

                Example 8<\/h3>\r\nSimplify [latex]6a^{4}+4a^{4}[\/latex].\r\n\r\n[reveal-answer q=\"840415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"840415\"]Notice that both terms have a number multiplied by [latex]a^{4}[\/latex]. This makes them like terms.\r\n

                [latex]6a^{4}+4a^{4}[\/latex]<\/p>\r\nCombine the coefficients, 6 and 4.\r\n

                [latex]\\left(6+4\\right)\\left(a^{4}\\right)[\/latex]<\/p>\r\nCalculate the sum.\r\n

                [latex]\\left(10\\right)\\left(a^{4}\\right)[\/latex]<\/p>\r\nWrite the sum as the new coefficient.\r\n

                Answer<\/h4>\r\n[latex]6a^{4}+4a^{4}=10a^{4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you have a polynomial with more terms, you have to be careful that you combine only<\/i> like terms.<\/i> If two terms are not like terms, you can\u2019t combine them.\r\n
                \r\n

                Example 9<\/h3>\r\nSimplify [latex]3x^{2}+3x+x+1+5x[\/latex]\r\n\r\n[reveal-answer q=\"731804\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"731804\"]First identify which terms are like terms<\/i>: only [latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex]\u00a0are like terms.\r\n\r\n[latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex] are like terms.\r\n\r\nUse the commutative and associative properties to group the like terms together.\r\n

                [latex]\\begin{array}{l}3x^{2}+3x+x+1+5x\\\\3x^{2}+\\left(3x+x+5x\\right)+1\\end{array}[\/latex]<\/p>\r\nAdd the coefficients of the like terms. Remember that the coefficient of x<\/em> is [latex]1\\left(x=1x\\right)[\/latex].\r\n

                [latex]\\begin{array}{l}3x^{2}+\\left(3+1+5\\right)x+1\\\\3x^{2}+\\left(9\\right)x+1\\end{array}[\/latex]<\/p>\r\nWrite the sum as the new coefficient.\r\n

                Answer<\/h4>\r\n[latex]3x^{2}+3x+x+1+5x=3x^{2}+9x+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/1epjbVO_qU4\r\n

                Domain of a polynomial function<\/h2>\r\nIn Section 3.2, we introduced domain as the set of input values for a function. In the context of a function given by a formula, it is helpful to expand upon this definition, where\u00a0domain<\/strong> is the set of input values for a function which produce valid input values. It follows given a function [latex]f(x)[\/latex], the domain of the function is the set of [latex]x[\/latex]-values that produce valid outputs.\r\n
                \r\n

                Example 10<\/h3>\r\nLet [latex]f(x)=-2x^2+7[\/latex].\r\n

                A.\u00a0 Compute [latex]f(-3)[\/latex], [latex]f(0)[\/latex], and [latex]f(2)[\/latex].\u00a0 If the answer is undefined, state this.<\/p>\r\n

                B.\u00a0 Are each of the values, [latex]-3[\/latex], [latex]0[\/latex], and [latex]2[\/latex], in the domain of [latex]f(x)[\/latex]?<\/p>\r\n

                C.\u00a0 Give the domain in interval notation.<\/p>\r\n[reveal-answer q=\"95795\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95795\"]\r\n

                A.\u00a0 As we did earlier in this section, we will substitute each of the given values in for [latex]x[\/latex].<\/p>\r\n

                [latex]f(-3)=-2(-3)^2+7=-2(9)+7=-18+7=-11[\/latex]<\/p>\r\n

                [latex]f(0)=-2(0)^2+7=-2(0)+7=0+7=7[\/latex]<\/p>\r\n

                [latex]f(2)=-2(2)^2+7=-2(4)+7=-8+7=-1[\/latex]<\/p>\r\n

                B.\u00a0 For each of the inputs [latex]-3,0,2[\/latex], we obtained valid outputs (as opposed to undefined). We conclude that all three values are in the domain of the function.<\/p>\r\n

                C.\u00a0 There was nothing particularly special about the inputs used in this problem. No matter what value we substitute for [latex]x[\/latex] in this function, we would obtain a valid input. So, we determine that the domain is \"all real numbers.\"\u00a0 In interval notation, we have the following domain:<\/p>\r\n

                Domain: [latex](-\\infty,\\infty)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf we consider the question of domain further, we realize that there was also nothing special about the function given in the previous example in terms of producing valid outputs. It follows that the domain of\u00a0every\u00a0<\/em>polynomial is \"all real numbers.\"\r\n

                Summary<\/h2>\r\nPolynomials are algebraic expressions that contain any number of terms combined by using addition or subtraction. A term is a number, a variable, or a product of a number and one or more variables with exponents. Like terms (same variable or variables raised to the same power) can be combined to simplify a polynomial. The polynomials can be evaluated by substituting a given value of the variable into each<\/i> instance of the variable, then using order of operations to complete the calculations. Lastly, we examined the domain of polynomial functions, revealing a domain of \"all real numbers\" for all polynomials.","rendered":"
                \n

                section 5.4 Learning Objectives<\/h3>\n

                5.4: Introduction to Polynomials<\/strong><\/p>\n

                  \n
                • Determine if an expression is a polynomial<\/li>\n
                • Identify the characteristics of a polynomial<\/li>\n
                • Determine if a polynomial is a monomial, binomial, or trinomial<\/li>\n
                • Evaluate a polynomial for a specified value<\/li>\n
                • Simplify polynomials by combining like terms<\/li>\n
                • Determine the domain of a polynomial function<\/li>\n<\/ul>\n<\/div>\n

                   <\/p>\n

                  Polynomials are algebraic expressions that are created by combining numbers and variables using arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation. You can create a polynomial<\/b> by adding or subtracting terms. Polynomials are very useful in applications from science and engineering to business. You may see a resemblance between expressions, which we have been studying in this course, and polynomials. \u00a0Polynomials are a special sub-group of mathematical expressions and equations.<\/p>\n

                  Determine if an expression is a polynomial<\/h2>\n

                  The following table is intended to help you tell the difference between what is a polynomial and what is not.<\/p>\n\n\n\n\n\n\n\n
                  IS a Polynomial<\/strong><\/td>\nIs NOT a Polynomial<\/strong><\/td>\nBecause<\/strong><\/td>\n<\/tr>\n<\/thead>\n
                  [latex]2x^2-\\frac{1}{2}x -9[\/latex]<\/td>\n[latex]\\frac{2}{x^{2}}+x[\/latex]<\/td>\nPolynomials only have variables in the numerator<\/td>\n<\/tr>\n
                  [latex]\\frac{y}{4}-y^3[\/latex]<\/td>\n[latex]\\frac{2}{y}+4[\/latex]<\/td>\nPolynomials only have variables in the numerator<\/td>\n<\/tr>\n
                  [latex]\\sqrt{12}\\left(a\\right)+9[\/latex]<\/td>\n\u00a0[latex]\\sqrt{a}+7[\/latex]<\/td>\n\u00a0Variables under a root are not allowed in polynomials<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  Identify the characteristics of a polynomial<\/h2>\n

                  The basic building block of a polynomial is a monomial<\/b>. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the coefficient<\/b>.<\/p>\n

                  Examples of monomials:<\/p>\n

                    \n
                  • number: [latex]{2}[\/latex]<\/li>\n
                  • variable: [latex]{x}[\/latex]<\/li>\n
                  • product of number and variable: [latex]{2x}[\/latex]<\/li>\n
                  • product of number and variable with an exponent: [latex]{2x}^{3}[\/latex]<\/li>\n<\/ul>\n

                    \"The<\/p>\n

                    The coefficient can be any real number, including 0. The exponent of the variable must be a whole number\u20140, 1, 2, 3, and so on. A monomial cannot have a variable in the denominator or a negative exponent.<\/p>\n

                    For a monomial in one variable, the value of the exponent is called the degree<\/b> of the monomial. Based on our exponent rules in Section 5.1, recall that [latex]x^{0}=1[\/latex].\u00a0 So, a monomial with no variable actually has a degree of 0.\u00a0 For example, we could rewrite the monomial 3 as [latex]3x^{0}[\/latex].<\/p>\n

                    \n

                    Example 1<\/h3>\n

                    Identify the coefficient, variable, and degree\u00a0of the variable for the following monomial terms:
                    \n1) 9
                    \n2) [latex]x[\/latex]
                    \n3) [latex]\\displaystyle \\frac{3}{5}{{k}^{8}}[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n1)
                    \n9 is a constant so there is no coefficient or variable. Since there is no variable, we consider the degree to be 0.<\/p>\n

                    2)
                    \nThe variable is [latex]x[\/latex].<\/p>\n

                    The exponent of [latex]x[\/latex] is 1 since [latex]x=x^{1}[\/latex].\u00a0So, the degree is 1.<\/p>\n

                    The coefficient of [latex]x[\/latex] is 1 since [latex]x=1x^{1}[\/latex].<\/p>\n

                    3)
                    \nThe variable is [latex]k[\/latex] .<\/p>\n

                    The exponent of [latex]k[\/latex] is 8, so the degree is 8.<\/p>\n

                    The coefficient of [latex]k^{8}[\/latex]\u00a0is [latex]\\displaystyle \\frac{3}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                    A polynomial is a monomial or the sum or difference of two or more monomials. Each monomial is called a term <\/b>of the polynomial.<\/p>\n

                    The word \u201cpolynomial\u201d has the prefix, \u201cpoly,\u201d which means many. However, the word polynomial can be used for all numbers of terms, including only one term.<\/p>\n

                    Because the exponent of the variable must be a whole number, monomials and polynomials cannot have a variable in the denominator.<\/p>\n

                    Polynomials can be classified by the degree of the polynomial. The degree of a polynomial<\/strong> is the degree of its highest-degree term. The coefficient of the highest degree term is called the leading coefficient<\/strong>.\u00a0\u00a0So the degree of [latex]2x^{3}+3x^{2}+8x+5[\/latex] is 3 and the leading coefficient is 2.<\/p>\n

                    A polynomial is said to be written in standard form (or “descending order”) when the terms are arranged from the highest-degree to the lowest degree. When it is written in standard form it is easy to determine the degree of the polynomial.<\/p>\n

                    If a polynomial contains a term with no variable, it is called the\u00a0constant term<\/strong>. If the polynomial is in standard form, the constant term appears at the end. If no constant term is explicitly given, then the constant term is 0.<\/p>\n

                    \n

                    terminology<\/h3>\n
                      \n
                    • Polynomial\u00a0<\/strong>– a sum of monomials, each called a term of the polynomial<\/li>\n
                    • Degree of a Polynomial<\/strong> – the degree of its highest-degree term<\/li>\n
                    • Leading Coefficient\u00a0<\/strong>– the coefficient of the highest-degree term<\/li>\n
                    • Constant Term\u00a0<\/strong>– the term with no variable (if no such term is written, it is 0)<\/li>\n<\/ul>\n<\/div>\n
                      \n

                      Example 2<\/h3>\n

                      For each polynomial, determine the number of terms, the degree of the polynomial, the leading coefficient, and the constant term.<\/p>\n

                        \n
                      1. [latex]\\hspace{.05in} 7x^3-4x^2+5x+8[\/latex]<\/li>\n
                      2. [latex]\\hspace{.05in}-3x^9+2x[\/latex]<\/li>\n
                      3. [latex]\\hspace{.05in}x^{4}-x-1[\/latex]<\/li>\n
                      4. [latex]\\hspace{.05in}13-x^2+2x+\\dfrac{3}{4}x^{5}+x^3[\/latex]<\/li>\n<\/ol>\n
                        Show Solution<\/span><\/p>\n
                        \n

                        1.\u00a0 [latex]\\hspace{.05in} 7x^3-4x^2+5x+8[\/latex]<\/p>\n

                        The polynomial has 4 terms (the 4 terms being [latex]7x^3[\/latex], [latex]-4x^2[\/latex], [latex]5x[\/latex], and [latex]8[\/latex]).<\/p>\n

                        Since the largest degree is [latex]3[\/latex], the degree of the polynomial is [latex]3[\/latex].<\/p>\n

                        The coefficient of the third degree term is [latex]7[\/latex], so the leading coefficient is [latex]7[\/latex].<\/p>\n

                        The constant term is [latex]8[\/latex], as this is the term that does not contain a variable.<\/p>\n

                        2.\u00a0 [latex]\\hspace{.05in}-3x^9+2x[\/latex]<\/p>\n

                        The polynomial has 2 terms.<\/p>\n

                        The degree of the polynomial is [latex]9[\/latex].<\/p>\n

                        The leading coefficient is [latex]-3[\/latex].<\/p>\n

                        Since no constant term is included, the constant term is [latex]0[\/latex].<\/p>\n

                        3.\u00a0 [latex]\\hspace{.05in}x^4-x-1[\/latex]<\/p>\n

                        The polynomial has 3 terms.<\/p>\n

                        The degree of the polynomial is [latex]4[\/latex].<\/p>\n

                        Since[latex]x^4=1x^4[\/latex], the leading coefficient is [latex]1[\/latex].<\/p>\n

                        The constant term is [latex]-1[\/latex] (be sure to include the negative sign with the term).<\/p>\n

                        4.\u00a0 [latex]\\hspace{.05in}13-x^2+2x+\\dfrac{3}{4}x^5+x^3[\/latex]<\/p>\n

                        Note that the polynomial is not in standard form. While we could still identify the relevant features, you might find it easier to first rewrite the polynomial in descending order as<\/p>\n

                        [latex]\\dfrac{3}{4}x^5+x^3-x^2+2x+13[\/latex].<\/p>\n

                        The polynomial has 5 terms.<\/p>\n

                        The degree of the polynomial is [latex]5[\/latex].<\/p>\n

                        The leading coefficient is [latex]\\dfrac{3}{4}[\/latex].<\/p>\n

                        The constant term is [latex]13[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                        Determine if a polynomial is a monomial, binomial, or trinomial<\/h2>\n

                        Some polynomials have specific names indicated by their prefix.<\/p>\n

                          \n
                        • Monomial<\/b>\u2014is a polynomial with exactly one term (\u201cmono\u201d\u2014means one)<\/li>\n
                        • Binomial<\/b>\u2014is a polynomial with exactly two terms (\u201cbi\u201d\u2014means two)<\/li>\n
                        • Trinomial<\/b>\u2014is a polynomial with exactly three terms (\u201ctri\u201d\u2014means three)<\/li>\n<\/ul>\n

                          The table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.<\/p>\n\n\n\n\n\n\n\n\n
                          Monomials<\/b><\/td>\nBinomials<\/b><\/td>\nTrinomials<\/b><\/td>\nOther Polynomials<\/b><\/td>\n<\/tr>\n<\/thead>\n
                          15<\/td>\n[latex]3y+13[\/latex]<\/td>\n[latex]x^{3}-x^{2}+1[\/latex]<\/td>\n[latex]5x^{4}+3x^{3}-6x^{2}+2x[\/latex]<\/td>\n<\/tr>\n
                          [latex]\\displaystyle \\frac{1}{2}x[\/latex]<\/td>\n[latex]4p-7[\/latex]<\/td>\n[latex]3x^{2}+2x-9[\/latex]<\/td>\n[latex]\\frac{1}{3}x^{5}-2x^{4}+\\frac{2}{9}x^{3}-x^{2}+4x-\\frac{5}{6}[\/latex]<\/td>\n<\/tr>\n
                          [latex]-4y^{3}[\/latex]<\/td>\n[latex]3x^{2}+\\frac{5}{8}x[\/latex]<\/td>\n[latex]3y^{3}+y^{2}-2[\/latex]<\/td>\n[latex]3t^{3}-3t^{2}-3t-3[\/latex]<\/td>\n<\/tr>\n
                          [latex]16n^{4}[\/latex]<\/td>\n[latex]14y^{3}+3y[\/latex]<\/td>\n[latex]a^{7}+2a^{5}-3a^{3}[\/latex]<\/td>\n[latex]q^{7}+2q^{5}-3q^{3}+q[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                          When the coefficient of a polynomial term is 0, you usually do not write the term at all (because 0 times anything is 0, and adding 0 doesn\u2019t change the value). The last binomial above could be written as a trinomial, [latex]14y^{3}+0y^{2}+3y[\/latex].<\/p>\n

                          \n

                          Example 3<\/h3>\n

                          For the following expressions, determine whether they are a polynomial. If so, categorize them as a monomial, binomial, or trinomial.<\/p>\n

                            \n
                          1. [latex]\\frac{x-3}{1-x}+x^2[\/latex]<\/li>\n
                          2. [latex]t^2+2t-3[\/latex]<\/li>\n
                          3. [latex]x^3+\\frac{x}{8}[\/latex]<\/li>\n
                          4. [latex]\\frac{\\sqrt{y}}{2}-y-1[\/latex]<\/li>\n<\/ol>\n
                            Show Solution<\/span><\/p>\n
                            \n
                              \n
                            1. [latex]\\frac{x-3}{1-x}+x^2[\/latex] is not a polynomial because it violates the rule that polynomials cannot have variables in the denominator of a fraction.<\/li>\n
                            2. [latex]t^2+2t-3[\/latex] is a polynomial because it is an expression whose terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. \u00a0There are three terms in this polynomial so it is a trinomial.<\/li>\n
                            3. [latex]x^3+\\frac{x}{8}[\/latex]is a polynomial because it is an expression whose terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. \u00a0There are two terms in this polynomial so it is a binomial.<\/li>\n
                            4. [latex]\\frac{\\sqrt{y}}{2}-y-1[\/latex]\u00a0is not a polynomial because it violates the rule that polynomials cannot have variables\u00a0under a root.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n

                              In the following video, you will be shown more examples of how to identify and categorize polynomials.<\/p>\n