{"id":1385,"date":"2015-11-12T18:35:29","date_gmt":"2015-11-12T18:35:29","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1385"},"modified":"2017-03-31T22:56:22","modified_gmt":"2017-03-31T22:56:22","slug":"find-zeros-of-a-polynomial-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/find-zeros-of-a-polynomial-function\/","title":{"raw":"Find zeros of a polynomial function","rendered":"Find zeros of a polynomial function"},"content":{"raw":"

The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division<\/span> repeatedly to determine all of the zeros<\/span> of a polynomial function.<\/p>\r\n\r\n

\r\n

How To: Given a polynomial function [latex]f[\/latex], use synthetic division to find its zeros.<\/h3>\r\n
  1. Use the Rational Zero Theorem to list all possible rational zeros of the function.<\/li>\r\n\t
  2. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.<\/li>\r\n\t
  3. Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic.<\/li>\r\n\t
  4. Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the quadratic formula.<\/li>\r\n<\/ol><\/div>\r\n
    \r\n
    \r\n
    \r\n

    Example 5: Finding the Zeros of a Polynomial Function with Repeated Real Zeros<\/h3>\r\n

    Find the zeros of [latex]f\\left(x\\right)=4{x}^{3}-3x - 1[\/latex].<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Solution<\/h3>\r\n

    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.<\/p>\r\n\r\n

    [latex]\\begin{cases}\\frac{p}{q}=\\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}}\\hfill \\\\ \\text{ }=\\frac{\\text{factor of -1}}{\\text{factor of 4}}\\hfill \\end{cases}[\/latex]<\/div>\r\n

    The 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 0. Let\u2019s begin with 1.<\/p>\r\n\r\n

    \"Synthetic<\/a><\/div>\r\n

    Dividing by [latex]\\left(x - 1\\right)[\/latex]\u00a0gives a remainder of 0, so 1 is a zero of the function. The polynomial can be written as<\/p>\r\n\r\n

    [latex]\\left(x - 1\\right)\\left(4{x}^{2}+4x+1\\right)[\/latex].<\/div>\r\n

    The quadratic is a perfect square. [latex]f\\left(x\\right)[\/latex]\u00a0can be written as<\/p>\r\n\r\n

    [latex]\\left(x - 1\\right){\\left(2x+1\\right)}^{2}[\/latex].<\/div>\r\n

    We already know that 1 is a zero. The other zero will have a multiplicity of 2 because the factor is squared. To find the other zero, we can set the factor equal to 0.<\/p>\r\n\r\n

    [latex]\\begin{cases}2x+1=0\\hfill \\\\ \\text{ }x=-\\frac{1}{2}\\hfill \\end{cases}[\/latex]<\/div>\r\n

    The zeros of the function are 1 and [latex]-\\frac{1}{2}[\/latex] with multiplicity 2.<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Analysis of the Solution<\/h3>\r\nLook at the graph of the function f<\/em>\u00a0in Figure 1. Notice, at [latex]x=-0.5[\/latex], the graph bounces off the x<\/em>-axis, indicating the even multiplicity (2,4,6\u2026) for the zero \u20130.5.\u00a0At [latex]x=1[\/latex], the graph crosses the x<\/em>-axis, indicating the odd multiplicity (1,3,5\u2026) for the zero [latex]x=1[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 1<\/b>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

    The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division<\/span> repeatedly to determine all of the zeros<\/span> of a polynomial function.<\/p>\n

    \n

    How To: Given a polynomial function [latex]f[\/latex], use synthetic division to find its zeros.<\/h3>\n
      \n
    1. Use the Rational Zero Theorem to list all possible rational zeros of the function.<\/li>\n
    2. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.<\/li>\n
    3. Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a quadratic.<\/li>\n
    4. Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the quadratic formula.<\/li>\n<\/ol>\n<\/div>\n
      \n
      \n
      \n

      Example 5: Finding the Zeros of a Polynomial Function with Repeated Real Zeros<\/h3>\n

      Find the zeros of [latex]f\\left(x\\right)=4{x}^{3}-3x - 1[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      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.<\/p>\n

      [latex]\\begin{cases}\\frac{p}{q}=\\frac{\\text{factor of constant term}}{\\text{factor of leading coefficient}}\\hfill \\\\ \\text{ }=\\frac{\\text{factor of -1}}{\\text{factor of 4}}\\hfill \\end{cases}[\/latex]<\/div>\n

      The 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].
      \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 0. Let\u2019s begin with 1.<\/p>\n

      \"Synthetic<\/a><\/div>\n

      Dividing by [latex]\\left(x - 1\\right)[\/latex]\u00a0gives a remainder of 0, so 1 is a zero of the function. The polynomial can be written as<\/p>\n

      [latex]\\left(x - 1\\right)\\left(4{x}^{2}+4x+1\\right)[\/latex].<\/div>\n

      The quadratic is a perfect square. [latex]f\\left(x\\right)[\/latex]\u00a0can be written as<\/p>\n

      [latex]\\left(x - 1\\right){\\left(2x+1\\right)}^{2}[\/latex].<\/div>\n

      We already know that 1 is a zero. The other zero will have a multiplicity of 2 because the factor is squared. To find the other zero, we can set the factor equal to 0.<\/p>\n

      [latex]\\begin{cases}2x+1=0\\hfill \\\\ \\text{ }x=-\\frac{1}{2}\\hfill \\end{cases}[\/latex]<\/div>\n

      The zeros of the function are 1 and [latex]-\\frac{1}{2}[\/latex] with multiplicity 2.<\/p>\n<\/div>\n

      \n

      Analysis of the Solution<\/h3>\n

      Look at the graph of the function f<\/em>\u00a0in Figure 1. Notice, at [latex]x=-0.5[\/latex], the graph bounces off the x<\/em>-axis, indicating the even multiplicity (2,4,6\u2026) for the zero \u20130.5.\u00a0At [latex]x=1[\/latex], the graph crosses the x<\/em>-axis, indicating the odd multiplicity (1,3,5\u2026) for the zero [latex]x=1[\/latex].<\/p>\n

      \"Graph<\/p>\n

      Figure 1<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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