{"id":284,"date":"2015-09-18T20:35:55","date_gmt":"2015-09-18T20:35:55","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=284"},"modified":"2015-11-03T19:57:05","modified_gmt":"2015-11-03T19:57:05","slug":"identifying-the-degree-and-leading-coefficient-of-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/identifying-the-degree-and-leading-coefficient-of-polynomials\/","title":{"raw":"Identifying the Degree and Leading Coefficient of Polynomials","rendered":"Identifying the Degree and Leading Coefficient of Polynomials"},"content":{"raw":"The formula just found is an example of a polynomial<\/strong>, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as [latex]384\\pi [\/latex], is known as a coefficient<\/strong>. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product [latex]{a}_{i}{x}^{i}[\/latex], such as [latex]384\\pi w[\/latex], is a term of a polynomial<\/strong>. If a term does not contain a variable, it is called a constant<\/em>.\r\n\r\nA polynomial containing only one term, such as [latex]5{x}^{4}[\/latex], is called a monomial<\/strong>. A polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a binomial<\/strong>. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a trinomial<\/strong>.\r\n\r\nWe can find the degree<\/strong> of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term<\/strong> because it is usually written first. The coefficient of the leading term is called the leading coefficient<\/strong>. When a polynomial is written so that the powers are descending, we say that it is in standard form.\r\n\r\n\"A\r\n
\r\n

A General Note: Polynomials<\/h3>\r\nA polynomial<\/strong> is an expression that can be written in the form\r\n
[latex]{a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\r\nEach real number ai<\/sub><\/em>is called a coefficient<\/strong>. The number [latex]{a}_{0}[\/latex] that is not multiplied by a variable is called a constant<\/em>. Each product [latex]{a}_{i}{x}^{i}[\/latex] is a term of a polynomial<\/strong>. The highest power of the variable that occurs in the polynomial is called the degree<\/strong> of a polynomial. The leading term<\/strong> is the term with the highest power, and its coefficient is called the leading coefficient<\/strong>.\r\n\r\n<\/div>\r\n
\r\n

How To: Given a polynomial expression, identify the degree and leading coefficient.<\/h3>\r\n
    \r\n\t
  1. Find the highest power of x<\/em> to determine the degree.<\/li>\r\n\t
  2. Identify the term containing the highest power of x<\/em> to find the leading term.<\/li>\r\n\t
  3. Identify the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Example 1: Identifying the Degree and Leading Coefficient of a Polynomial<\/h3>\r\nFor the following polynomials, identify the degree, the leading term, and the leading coefficient.\r\n
      \r\n\t
    1. [latex]3+2{x}^{2}-4{x}^{3}\\\\[\/latex]<\/li>\r\n\t
    2. [latex]5{t}^{5}-2{t}^{3}+7t[\/latex]<\/li>\r\n\t
    3. [latex]6p-{p}^{3}-2[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Solution<\/h3>\r\n
        \r\n\t
      1. The highest power of x<\/em> is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\r\n\t
      2. The highest power of t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\r\n\t
      3. The highest power of p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n
        \r\n

        Try It 1<\/h3>\r\nIdentify the degree, leading term, and leading coefficient of the polynomial [latex]4{x}^{2}-{x}^{6}+2x - 6[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

        The formula just found is an example of a polynomial<\/strong>, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as [latex]384\\pi[\/latex], is known as a coefficient<\/strong>. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product [latex]{a}_{i}{x}^{i}[\/latex], such as [latex]384\\pi w[\/latex], is a term of a polynomial<\/strong>. If a term does not contain a variable, it is called a constant<\/em>.<\/p>\n

        A polynomial containing only one term, such as [latex]5{x}^{4}[\/latex], is called a monomial<\/strong>. A polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a binomial<\/strong>. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a trinomial<\/strong>.<\/p>\n

        We can find the degree<\/strong> of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term<\/strong> because it is usually written first. The coefficient of the leading term is called the leading coefficient<\/strong>. When a polynomial is written so that the powers are descending, we say that it is in standard form.<\/p>\n

        \"A<\/p>\n

        \n

        A General Note: Polynomials<\/h3>\n

        A polynomial<\/strong> is an expression that can be written in the form<\/p>\n

        [latex]{a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n

        Each real number ai<\/sub><\/em>is called a coefficient<\/strong>. The number [latex]{a}_{0}[\/latex] that is not multiplied by a variable is called a constant<\/em>. Each product [latex]{a}_{i}{x}^{i}[\/latex] is a term of a polynomial<\/strong>. The highest power of the variable that occurs in the polynomial is called the degree<\/strong> of a polynomial. The leading term<\/strong> is the term with the highest power, and its coefficient is called the leading coefficient<\/strong>.<\/p>\n<\/div>\n

        \n

        How To: Given a polynomial expression, identify the degree and leading coefficient.<\/h3>\n
          \n
        1. Find the highest power of x<\/em> to determine the degree.<\/li>\n
        2. Identify the term containing the highest power of x<\/em> to find the leading term.<\/li>\n
        3. Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n
          \n

          Example 1: Identifying the Degree and Leading Coefficient of a Polynomial<\/h3>\n

          For the following polynomials, identify the degree, the leading term, and the leading coefficient.<\/p>\n

            \n
          1. [latex]3+2{x}^{2}-4{x}^{3}\\\\[\/latex]<\/li>\n
          2. [latex]5{t}^{5}-2{t}^{3}+7t[\/latex]<\/li>\n
          3. [latex]6p-{p}^{3}-2[\/latex]<\/li>\n<\/ol>\n<\/div>\n
            \n

            Solution<\/h3>\n
              \n
            1. The highest power of x<\/em> is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\n
            2. The highest power of t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\n
            3. The highest power of p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\n<\/ol>\n<\/div>\n
              \n

              Try It 1<\/h3>\n

              Identify the degree, leading term, and leading coefficient of the polynomial [latex]4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\n

              Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

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