{"id":925,"date":"2025-07-16T18:20:46","date_gmt":"2025-07-16T18:20:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=925"},"modified":"2026-01-13T16:08:04","modified_gmt":"2026-01-13T16:08:04","slug":"graphs-of-polynomial-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-polynomial-functions-learn-it-2\/","title":{"raw":"Graphs of Polynomial Functions: Learn It 2","rendered":"Graphs of Polynomial Functions: Learn It 2"},"content":{"raw":"
\r\n

Using Factoring to Find Zeros of Polynomial Functions<\/h2>\r\nRecall that if [latex]f[\/latex] is a polynomial function, the values of [latex]x[\/latex] for which [latex]f(x) = 0[\/latex] are called zeros of [latex]f[\/latex]. If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.\r\n\r\nWe can use this method to find [latex]x[\/latex]-intercepts because at the [latex]x[\/latex]-intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases:\r\n
    \r\n \t
  1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.<\/li>\r\n \t
  2. The polynomial is given in factored form.<\/li>\r\n \t
  3. Technology is used to determine the intercepts.<\/li>\r\n<\/ol>\r\n
    How to: Given a polynomial function [latex]f[\/latex], find the [latex]x[\/latex]-intercepts by factoring.<\/strong>\r\n
      \r\n \t
    1. Set [latex]f(x) = 0[\/latex].<\/li>\r\n \t
    2. If the polynomial function is not given in factored form:\r\n
        \r\n \t
      • Factor out any common monomial factors.<\/li>\r\n \t
      • Factor any factorable binomials or trinomials.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
      • Set each factor equal to zero and solve to find the [latex]x[\/latex]-intercepts.<\/li>\r\n<\/ol>\r\n<\/section>
        Find the [latex]x[\/latex]-intercepts of [latex]f(x) = x^6 - 3x^4 + 2x^2[\/latex].[reveal-answer q=\"431492\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"431492\"]We can attempt to factor this polynomial to find solutions for [latex]f(x) = 0[\/latex].\r\n

        [latex] \\begin{array}{rcl} x^6 - 3x^4 + 2x^2 & = & 0 \\quad \\text{Factor out the greatest common factor.} \\\\ x^2 (x^4 - 3x^2 + 2) & = & 0 \\quad \\text{Factor the trinomial.} \\\\ x^2 (x^2 - 1)(x^2 - 2) & = & 0 \\quad \\text{Set each factor equal to zero.} \\end{array} [\/latex]<\/p>\r\n

        [latex] \\begin{array}{rcl} x^2 & = & 0 \\quad \\text{or} \\quad (x^2 - 1) = 0 \\quad \\text{or} \\quad (x^2 - 2) = 0 \\\\ x & = & 0 \\quad \\quad \\quad \\quad \\quad \\quad x^2 = 1 \\quad \\quad \\quad x^2 = 2 \\\\ x & = & 0 \\quad \\quad \\quad \\quad \\quad \\quad x = \\pm 1 \\quad \\quad x = \\pm \\sqrt{2} \\end{array} [\/latex]<\/p>\r\nThis gives us five [latex]x[\/latex]-intercepts: [latex](0, 0)[\/latex], [latex](1, 0)[\/latex], [latex](-1, 0)[\/latex], [latex](\\sqrt{2}, 0)[\/latex], and [latex](-\\sqrt{2}, 0)[\/latex]. We can see that this is an even function because it is symmetric about the y-axis.\r\n\r\n\"This\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

        Find the\u00a0[latex]y[\/latex]- and\u00a0[latex]x[\/latex]-intercepts of [latex]g(x) = (x-2)^2(2x+3)[\/latex].[reveal-answer q=\"133872\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"133872\"] We can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown below.\"Graph[\/hidden-answer]<\/section>
        Find the [latex]x[\/latex]-intercepts of [latex]h(x) = x^3+4x^2+x-6[\/latex].[reveal-answer q=\"75721\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"75721\"]\r\n

        This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.<\/p>\r\n

        Looking at the graph of this function, as shown below, it appears that there are\u00a0[latex]x[\/latex]-intercepts at [latex]x= -3, -2, \\text{ and } 1[\/latex].<\/p>\r\n\"Graph\r\n\r\nWe can check whether these are correct by substituting these values for [latex]x[\/latex] and verifying that\r\n

        [latex] h(-3) = h(-2) = h(1) = 0 [\/latex]<\/p>\r\nSince [latex]h(x) = x^3 + 4x^2 + x - 6[\/latex], we have:\r\n

        [latex] \\begin{array}{rcl} h(-3) & = & (-3)^3 + 4(-3)^2 + (-3) - 6 = -27 + 36 - 3 - 6 = 0 \\\\ h(-2) & = & (-2)^3 + 4(-2)^2 + (-2) - 6 = -8 + 16 - 2 - 6 = 0 \\\\ h(1) & = & (1)^3 + 4(1)^2 + (1) - 6 = 1 + 4 + 1 - 6 = 0 \\end{array} [\/latex]<\/p>\r\nEach [latex]x[\/latex]-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.\r\n

        [latex] h(x) = x^3 + 4x^2 + x - 6 = (x + 3)(x + 2)(x - 1) [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

        We can find the [latex]y[\/latex]-intercept (or vertical intercept) of any function by finding [latex]f(0)[\/latex]. Just substitute [latex]x=0[\/latex] into the function's equation.<\/section><\/section>
        [ohm_question hide_question_numbers=1]318826[\/ohm_question]<\/section>","rendered":"
        \n

        Using Factoring to Find Zeros of Polynomial Functions<\/h2>\n

        Recall that if [latex]f[\/latex] is a polynomial function, the values of [latex]x[\/latex] for which [latex]f(x) = 0[\/latex] are called zeros of [latex]f[\/latex]. If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.<\/p>\n

        We can use this method to find [latex]x[\/latex]-intercepts because at the [latex]x[\/latex]-intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases:<\/p>\n

          \n
        1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.<\/li>\n
        2. The polynomial is given in factored form.<\/li>\n
        3. Technology is used to determine the intercepts.<\/li>\n<\/ol>\n
          How to: Given a polynomial function [latex]f[\/latex], find the [latex]x[\/latex]-intercepts by factoring.<\/strong><\/p>\n
            \n
          1. Set [latex]f(x) = 0[\/latex].<\/li>\n
          2. If the polynomial function is not given in factored form:\n
              \n
            • Factor out any common monomial factors.<\/li>\n
            • Factor any factorable binomials or trinomials.<\/li>\n<\/ul>\n<\/li>\n
            • Set each factor equal to zero and solve to find the [latex]x[\/latex]-intercepts.<\/li>\n<\/ol>\n<\/section>\n
              Find the [latex]x[\/latex]-intercepts of [latex]f(x) = x^6 - 3x^4 + 2x^2[\/latex].<\/p>\n