{"id":978,"date":"2025-07-17T22:29:09","date_gmt":"2025-07-17T22:29:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=978"},"modified":"2026-01-13T20:43:52","modified_gmt":"2026-01-13T20:43:52","slug":"zeros-of-polynomial-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/zeros-of-polynomial-functions-learn-it-2\/","title":{"raw":"Zeros of Polynomial Functions: Learn It 2","rendered":"Zeros of Polynomial Functions: Learn It 2"},"content":{"raw":"
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Factor Theorem<\/h2>\r\nThe Factor Theorem<\/strong> is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors.\r\n\r\n
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the factor theorem<\/h3>\r\nAccording to the factor theorem,<\/strong> [latex]k[\/latex] is a zero of [latex]f(x)[\/latex] if and only if [latex](x\u2212k)[\/latex] is a factor of [latex]f(x)[\/latex].\r\n\r\n<\/div>\r\n<\/section>Let's walk through the proof of the theorem.\r\n\r\n
Recall that the Division Algorithm.\r\n

[latex]f(x) = (x - k)q(x) + r[\/latex]<\/p>\r\nIf [latex]k[\/latex] is a zero, then the remainder [latex]r[\/latex] is [latex]f(k) = 0[\/latex] and [latex]f(x) = (x - k)q(x) + 0[\/latex] or [latex]f(x) = (x - k)q(x)[\/latex].\r\n\r\nNotice, written in this form, [latex]x - k[\/latex] is a factor of [latex]f(x)[\/latex]. We can conclude if [latex]k[\/latex] is a zero of [latex]f(x)[\/latex], then [latex]x - k[\/latex] is a factor of [latex]f(x)[\/latex].\r\n\r\nSimilarly, if [latex]x - k[\/latex] is a factor of [latex]f(x)[\/latex], then the remainder of the Division Algorithm [latex]f(x) = (x - k)q(x) + r[\/latex] is [latex]0[\/latex]. This tells us that [latex]k[\/latex] is a zero.\r\n\r\n<\/section>This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree [latex]n[\/latex] in the complex number system will have [latex]n[\/latex] zeros. We can use the Factor Theorem to completely factor a polynomial into the product of [latex]n[\/latex] factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.\r\n\r\n

How to: Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.<\/strong>\r\n
    \r\n \t
  1. Use synthetic division to divide the polynomial by [latex](x-k)[\/latex]<\/li>\r\n \t
  2. Confirm that the remainder is [latex]0[\/latex].<\/li>\r\n \t
  3. Write the polynomial as the product of [latex](x-k)[\/latex]\u00a0and the quadratic quotient.<\/li>\r\n \t
  4. If possible, factor the quadratic.<\/li>\r\n \t
  5. Write the polynomial as the product of factors.<\/li>\r\n<\/ol>\r\n<\/section>
    Show that [latex](x + 2)[\/latex] is a factor of [latex]x^3 - 6x^2 - x + 30[\/latex]. Find the remaining factors. Use the factors to determine the zeros of the polynomial.[reveal-answer q=\"981344\"]Show Solution[\/reveal-answer] [hidden-answer a=\"981344\"]We can use synthetic division to show that [latex](x + 2)[\/latex] is a factor of the polynomial.
    \"A<\/center>The remainder is zero, so [latex](x + 2)[\/latex] is a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:\r\n

    [latex](x + 2)(x^2 - 8x + 15)[\/latex]<\/p>\r\nWe can factor the quadratic factor to write the polynomial as\r\n

    [latex](x + 2)(x - 3)(x - 5)[\/latex]<\/p>\r\nBy the Factor Theorem, the zeros of [latex]x^3 - 6x^2 - x + 30[\/latex] are [latex]-2, 3,[\/latex] and [latex]5[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]318892[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]318893[\/ohm_question]<\/section>
    Find the zeros of [latex]f\\left(x\\right)=4{x}^{3}-3x - 1[\/latex] and graph the function.[reveal-answer q=\"571513\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"571513\"]The Rational Zero Theorem tells us that if [latex]\\frac{p}{q}[\/latex] is a zero of [latex]f\\left(x\\right)[\/latex], then p\u00a0<\/em>is a factor of \u20131 and\u00a0q<\/em>\u00a0is a factor of 4.\r\n

    [latex]\\begin{array}{l}\\frac{p}{q}=\\frac{\\text{Factors of the constant term}}{\\text{Factors of the leading coefficient}}\\hfill \\\\ \\text{}\\frac{p}{q}=\\frac{\\text{Factors of -1}}{\\text{Factors of 4}}\\hfill \\end{array}[\/latex]<\/p>\r\nThe factors of \u20131 are [latex]\\pm 1[\/latex]\u00a0and the factors of 4 are [latex]\\pm 1,\\pm 2[\/latex], and [latex]\\pm 4[\/latex]. The possible values for [latex]\\frac{p}{q}[\/latex] are [latex]\\pm 1,\\pm \\frac{1}{2}[\/latex], and [latex]\\pm \\frac{1}{4}[\/latex].\r\nThese are the possible rational zeros for the function. We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of [latex]0[\/latex]. Let\u2019s begin with [latex]1[\/latex].\r\n\r\n\"Synthetic<\/a>\r\n\r\nDividing by [latex]\\left(x - 1\\right)[\/latex]\u00a0gives a remainder of [latex]0[\/latex], so [latex]1[\/latex] is a zero of the function. The polynomial can be written as [latex]\\left(x - 1\\right)\\left(4{x}^{2}+4x+1\\right)[\/latex].\r\n\r\nThe quadratic is a perfect square. [latex]f\\left(x\\right)[\/latex]\u00a0can be written as [latex]\\left(x - 1\\right){\\left(2x+1\\right)}^{2}[\/latex].\r\n\r\nWe already know that [latex]1[\/latex] is a zero. The other zero will have a multiplicity of [latex]2[\/latex] because the factor is squared. To find the other zero, we can set the factor equal to [latex]0[\/latex].\r\n

    [latex]\\begin{array}{l}2x+1=0\\hfill \\\\ \\text{ }x=-\\frac{1}{2}\\hfill \\end{array}[\/latex]<\/p>\r\nThe zeros of the function are [latex]1[\/latex] and [latex]-\\frac{1}{2}[\/latex] with multiplicity [latex]2[\/latex].<\/strong>\r\n

    Graph of the function<\/h4>\r\n
      \r\n \t
    • At [latex]x=-0.5[\/latex], the graph touch (bounces off) the [latex]x[\/latex]-axis, indicating the even multiplicity for the zero [latex]\u20130.5[\/latex].<\/li>\r\n \t
    • At [latex]x=1[\/latex], the graph crosses the [latex]x[\/latex]-axis, indicating the odd multiplicity for the zero [latex]x=1[\/latex].<\/li>\r\n<\/ul>\r\n\"Graph\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section>
      [ohm_question hide_question_numbers=1]318897[\/ohm_question]<\/section>","rendered":"
      \n

      Factor Theorem<\/h2>\n

      The Factor Theorem<\/strong> is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors.<\/p>\n

      \n
      \n

      the factor theorem<\/h3>\n

      According to the factor theorem,<\/strong> [latex]k[\/latex] is a zero of [latex]f(x)[\/latex] if and only if [latex](x\u2212k)[\/latex] is a factor of [latex]f(x)[\/latex].<\/p>\n<\/div>\n<\/section>\n

      Let’s walk through the proof of the theorem.<\/p>\n

      Recall that the Division Algorithm.<\/p>\n

      [latex]f(x) = (x - k)q(x) + r[\/latex]<\/p>\n

      If [latex]k[\/latex] is a zero, then the remainder [latex]r[\/latex] is [latex]f(k) = 0[\/latex] and [latex]f(x) = (x - k)q(x) + 0[\/latex] or [latex]f(x) = (x - k)q(x)[\/latex].<\/p>\n

      Notice, written in this form, [latex]x - k[\/latex] is a factor of [latex]f(x)[\/latex]. We can conclude if [latex]k[\/latex] is a zero of [latex]f(x)[\/latex], then [latex]x - k[\/latex] is a factor of [latex]f(x)[\/latex].<\/p>\n

      Similarly, if [latex]x - k[\/latex] is a factor of [latex]f(x)[\/latex], then the remainder of the Division Algorithm [latex]f(x) = (x - k)q(x) + r[\/latex] is [latex]0[\/latex]. This tells us that [latex]k[\/latex] is a zero.<\/p>\n<\/section>\n

      This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree [latex]n[\/latex] in the complex number system will have [latex]n[\/latex] zeros. We can use the Factor Theorem to completely factor a polynomial into the product of [latex]n[\/latex] factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.<\/p>\n

      How to: Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.<\/strong><\/p>\n
        \n
      1. Use synthetic division to divide the polynomial by [latex](x-k)[\/latex]<\/li>\n
      2. Confirm that the remainder is [latex]0[\/latex].<\/li>\n
      3. Write the polynomial as the product of [latex](x-k)[\/latex]\u00a0and the quadratic quotient.<\/li>\n
      4. If possible, factor the quadratic.<\/li>\n
      5. Write the polynomial as the product of factors.<\/li>\n<\/ol>\n<\/section>\n
        Show that [latex](x + 2)[\/latex] is a factor of [latex]x^3 - 6x^2 - x + 30[\/latex]. Find the remaining factors. Use the factors to determine the zeros of the polynomial.<\/p>\n