{"id":917,"date":"2025-07-16T18:11:33","date_gmt":"2025-07-16T18:11:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=917"},"modified":"2026-03-06T21:22:15","modified_gmt":"2026-03-06T21:22:15","slug":"polynomial-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polynomial-functions-learn-it-3\/","title":{"raw":"Polynomial Functions: Learn It 3","rendered":"Polynomial Functions: Learn It 3"},"content":{"raw":"
\r\n

Identifying Intercepts of Factored Polynomial Functions<\/h2>\r\n

When polynomial functions are written in factored form, finding their intercepts becomes straightforward. The factored form reveals the zeros directly, while the y-intercept can be found through substitution. This page will teach you to identify both types of intercepts efficiently.<\/p>\r\n\r\n

\r\n

factored form of a polynomial<\/h3>\r\n

A polynomial function in factored form is written as:<\/p>\r\n

[latex]f(x) = a(x - r_1)(x - r_2)(x - r_3)\\cdots(x - r_n)[\/latex]<\/p>\r\n

where:<\/p>\r\n\r\n

    \r\n \t
  • [latex]a[\/latex] is the leading coefficient<\/li>\r\n \t
  • [latex]r_1, r_2, r_3, \\ldots, r_n[\/latex] are the zeros (roots) of the function<\/li>\r\n<\/ul>\r\n<\/section>
    \r\n

    [latex]f(x) = 2(x - 3)(x + 1)(x - 5)[\/latex]<\/p>\r\n

    This polynomial has zeros at [latex]x = 3[\/latex], [latex]x = -1[\/latex], and [latex]x = 5[\/latex].<\/p>\r\n[reveal-answer q=\"781576\"]Explanation[\/reveal-answer]\r\n[hidden-answer a=\"781576\"]To find the zeroes of a polynomial function we set the original expression equal to zero:\r\n\r\n[latex]f(x) = 2(x - 3)(x + 1)(x - 5)=0[\/latex]\r\n\r\nThen, by the zero product property<\/strong>, any one of the factors would have to be equal to [latex]0[\/latex] for the entire expression to be equal to zero.\r\n\r\n[latex]x-3=0[\/latex]\r\n\r\n[latex]x=3[\/latex] or\r\n\r\n[latex]x+1=0[\/latex]\r\n\r\n[latex]x=-1[\/latex] or\r\n\r\n[latex]x-5=0[\/latex]\r\n\r\n[latex]x=5[\/latex]\r\n\r\n[\/hidden-answer]x-intercepts from Factored Form\r\n\r\n<\/section>

    \r\n

    How to: <\/strong>find x-intercepts from factored form:<\/p>\r\n\r\n

      \r\n \t
    1. Set the function equal to zero: [latex]f(x) = 0[\/latex]<\/li>\r\n \t
    2. Use the Zero Product Property<\/li>\r\n \t
    3. Solve each factor equal to zero<\/li>\r\n \t
    4. The solutions are the x-intercepts<\/li>\r\n<\/ol>\r\n<\/section>
      [ohm_question hide_question_numbers=1]317668[\/ohm_question]<\/section>
      \r\n

      Find the x-intercepts of [latex]h(x) = -2(x - 5)(x + 3)(x - 2)[\/latex].<\/p>\r\n

      Solution:<\/strong><\/p>\r\n

      Step 1: Set the function equal to zero [latex]-2(x - 5)(x + 3)(x - 2) = 0[\/latex]<\/p>\r\n

      Step 2: Apply the Zero Product Property Since [latex]-2 \\neq 0[\/latex], we focus on the factors containing [latex]x[\/latex]:<\/p>\r\n\r\n

        \r\n \t
      • [latex]x - 5 = 0[\/latex] or<\/strong><\/li>\r\n \t
      • [latex]x + 3 = 0[\/latex] or<\/strong><\/li>\r\n \t
      • [latex]x - 2 = 0[\/latex]<\/li>\r\n<\/ul>\r\n

        Step 3: Solve each equation<\/p>\r\n\r\n

          \r\n \t
        • [latex]x - 5 = 0 \\Rightarrow x = 5[\/latex]<\/li>\r\n \t
        • [latex]x + 3 = 0 \\Rightarrow x = -3[\/latex]<\/li>\r\n \t
        • [latex]x - 2 = 0 \\Rightarrow x = 2[\/latex]<\/li>\r\n<\/ul>\r\n

          Answer:<\/strong> The x-intercepts are [latex]x = 5[\/latex], [latex]x = -3[\/latex], and [latex]x = 2[\/latex].<\/p>\r\n\r\n<\/section>

          The leading coefficient affects the shape and direction of the graph but does not change the x-intercepts.<\/section>
          \r\n

          multiplicity<\/h3>\r\nWhen a factor appears multiple times, we say it has multiplicity greater than 1.\r\n\r\n<\/section>
          \r\n

          Find the x-intercepts of [latex]f(x) = (x - 2)^2(x + 1)(x - 3)[\/latex].<\/p>\r\n

          Solution:<\/strong><\/p>\r\n

          Step 1: Identify the factors<\/p>\r\n\r\n

            \r\n \t
          • [latex](x - 2)^2[\/latex] - factor [latex](x - 2)[\/latex] with multiplicity 2<\/li>\r\n \t
          • [latex](x + 1)[\/latex] - factor [latex](x + 1)[\/latex] with multiplicity 1<\/li>\r\n \t
          • [latex](x - 3)[\/latex] - factor [latex](x - 3)[\/latex] with multiplicity 1<\/li>\r\n<\/ul>\r\n

            Step 2: Find the zeros<\/p>\r\n\r\n

              \r\n \t
            • [latex]x - 2 = 0 \\Rightarrow x = 2[\/latex] (multiplicity 2)<\/li>\r\n \t
            • [latex]x + 1 = 0 \\Rightarrow x = -1[\/latex] (multiplicity 1)<\/li>\r\n \t
            • [latex]x - 3 = 0 \\Rightarrow x = 3[\/latex] (multiplicity 1)<\/li>\r\n<\/ul>\r\n

              Answer:<\/strong> The x-intercepts are [latex]x = 2[\/latex] (multiplicity 2), [latex]x = -1[\/latex], and [latex]x = 3[\/latex].<\/p>\r\n\r\n<\/section>\r\n

              Finding y-intercepts<\/h2>\r\n
              The y-intercept<\/strong> is the point where the graph crosses the y-axis, which occurs when [latex]x = 0[\/latex].<\/section>
              \r\n

              How to:<\/strong>\u00a0find the y-intercept:<\/p>\r\n\r\n

                \r\n \t
              1. Substitute [latex]x = 0[\/latex] into the function<\/li>\r\n \t
              2. Evaluate the expression<\/li>\r\n \t
              3. The result is the y-coordinate of the y-intercept<\/li>\r\n<\/ol>\r\n<\/section>
                \r\n

                Find the y-intercept of [latex]f(x) = 2(x - 3)(x + 1)(x - 5)[\/latex].<\/p>\r\n

                [reveal-answer q=\"598239\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"598239\"]<\/p>\r\n

                Substitute [latex]x = 0[\/latex]\r\n\r\n[latex]\\begin{align} f(0) &= 2(0 - 3)(0 + 1)(0 - 5)\\\\ &= 2(-3)(1)(-5)\\\\ &= 2(-3)(-5)\\\\ &= 2(15)\\\\ &= 30 \\end{align}[\/latex]<\/p>\r\n

                Answer:<\/strong> The y-intercept is [latex](0, 30)[\/latex].<\/p>\r\n

                [\/hidden-answer]<\/p>\r\n\r\n<\/section>

                \r\n

                [ohm_question hide_question_numbers=1]317669[\/ohm_question]<\/p>\r\n\r\n<\/section><\/div>","rendered":"

                \n

                Identifying Intercepts of Factored Polynomial Functions<\/h2>\n

                When polynomial functions are written in factored form, finding their intercepts becomes straightforward. The factored form reveals the zeros directly, while the y-intercept can be found through substitution. This page will teach you to identify both types of intercepts efficiently.<\/p>\n

                \n

                factored form of a polynomial<\/h3>\n

                A polynomial function in factored form is written as:<\/p>\n

                [latex]f(x) = a(x - r_1)(x - r_2)(x - r_3)\\cdots(x - r_n)[\/latex]<\/p>\n

                where:<\/p>\n

                  \n
                • [latex]a[\/latex] is the leading coefficient<\/li>\n
                • [latex]r_1, r_2, r_3, \\ldots, r_n[\/latex] are the zeros (roots) of the function<\/li>\n<\/ul>\n<\/section>\n
                  \n

                  [latex]f(x) = 2(x - 3)(x + 1)(x - 5)[\/latex]<\/p>\n

                  This polynomial has zeros at [latex]x = 3[\/latex], [latex]x = -1[\/latex], and [latex]x = 5[\/latex].<\/p>\n