{"id":1902,"date":"2024-06-24T19:19:12","date_gmt":"2024-06-24T19:19:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1902"},"modified":"2025-08-15T02:18:57","modified_gmt":"2025-08-15T02:18:57","slug":"introduction-to-power-and-polynomial-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-power-and-polynomial-functions-learn-it-4\/","title":{"raw":"Introduction to Power and Polynomial Functions: Learn It 4","rendered":"Introduction to Power and Polynomial Functions: Learn It 4"},"content":{"raw":"

Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h2>\r\nBecause of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order based on the power on the variable. This is called writing a polynomial in general or standard form.\r\n\r\n
\r\n

terminology of a polynomial function<\/h3>\r\n
    \r\n \t
  • The degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.<\/li>\r\n \t
  • The leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree.<\/li>\r\n \t
  • The leading coefficient<\/strong> is the coefficient of the leading term.<\/li>\r\n<\/ul>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Diagram Diagram to show the components of the leading term[\/caption]\r\n\r\n<\/section>
    How To: Given a polynomial function, identify the degree and leading coefficient<\/strong>\r\n
      \r\n \t
    1. Find the highest power of [latex]x[\/latex]to determine the degree of the function.<\/li>\r\n \t
    2. Identify the term containing the highest power of [latex]x[\/latex]to find the leading term.<\/li>\r\n \t
    3. The leading coefficient is the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/section>
      Identify the degree, leading term, and leading coefficient of the following polynomial functions.\r\n

      [latex]\\begin{array}{l} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"632394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"632394\"]\r\n\r\nFor the function [latex]f\\left(x\\right)[\/latex], the highest power of [latex]x[\/latex]\u00a0is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]\u20134[\/latex].\r\n\r\nFor the function [latex]g\\left(t\\right)[\/latex], the highest power of [latex]t[\/latex]\u00a0is [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].\r\n\r\nFor the function [latex]h\\left(p\\right)[\/latex], the highest power of [latex]p[\/latex]\u00a0is [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]\u20131[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

      [ohm2_question hide_question_numbers=1]24602[\/ohm2_question]<\/section>
      [ohm2_question hide_question_numbers=1]24603[\/ohm2_question]<\/section>","rendered":"

      Degree and Leading Coefficient\u00a0of a Polynomial Function<\/h2>\n

      Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order based on the power on the variable. This is called writing a polynomial in general or standard form.<\/p>\n

      \n

      terminology of a polynomial function<\/h3>\n
        \n
      • The degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form.<\/li>\n
      • The leading term<\/strong> is the term containing the variable with the highest power, also called the term with the highest degree.<\/li>\n
      • The leading coefficient<\/strong> is the coefficient of the leading term.<\/li>\n<\/ul>\n
        \"Diagram
        Diagram to show the components of the leading term<\/figcaption><\/figure>\n<\/section>\n
        How To: Given a polynomial function, identify the degree and leading coefficient<\/strong><\/p>\n
          \n
        1. Find the highest power of [latex]x[\/latex]to determine the degree of the function.<\/li>\n
        2. Identify the term containing the highest power of [latex]x[\/latex]to find the leading term.<\/li>\n
        3. The leading coefficient is the coefficient of the leading term.<\/li>\n<\/ol>\n<\/section>\n
          Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n

          [latex]\\begin{array}{l} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/p>\n