{"id":1380,"date":"2015-11-12T18:35:30","date_gmt":"2015-11-12T18:35:30","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1380"},"modified":"2017-03-31T22:52:43","modified_gmt":"2017-03-31T22:52:43","slug":"use-the-factor-theorem-to-solve-a-polynomial-equation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/use-the-factor-theorem-to-solve-a-polynomial-equation\/","title":{"raw":"Use the Factor Theorem to solve a polynomial equation","rendered":"Use the Factor Theorem to solve a polynomial equation"},"content":{"raw":"

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

[latex]f\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)+r[\/latex].<\/div>\r\n

If k<\/em>\u00a0is a zero, then the remainder r<\/em>\u00a0is [latex]f\\left(k\\right)=0[\/latex]\u00a0and [latex]f\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)+0[\/latex]\u00a0or [latex]f\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)[\/latex].<\/p>\r\n

Notice, written in this form, x<\/em>\u00a0\u2013\u00a0k<\/em> is a factor of [latex]f\\left(x\\right)[\/latex]. We can conclude if k\u00a0<\/em>is a zero of [latex]f\\left(x\\right)[\/latex], then [latex]x-k[\/latex] is a factor of [latex]f\\left(x\\right)[\/latex].<\/p>\r\n

Similarly, if [latex]x-k[\/latex]\u00a0is a factor of [latex]f\\left(x\\right)[\/latex],\u00a0then the remainder of the Division Algorithm [latex]f\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)+r[\/latex]\u00a0is 0. This tells us that k<\/em>\u00a0is a zero.<\/p>\r\n

This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree n<\/em>\u00a0in the complex number system will have n<\/em>\u00a0zeros. We can use the Factor Theorem to completely factor a polynomial into the product of n<\/em>\u00a0factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.<\/p>\r\n\r\n

\r\n

A General Note: The Factor Theorem<\/h3>\r\n

According to the Factor Theorem<\/strong>, k<\/em>\u00a0is a zero of [latex]f\\left(x\\right)[\/latex]\u00a0if and only if [latex]\\left(x-k\\right)[\/latex]\u00a0is a factor of [latex]f\\left(x\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n

\r\n

How To: Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.\r\n<\/strong><\/h3>\r\n
  1. Use synthetic division to divide the polynomial by [latex]\\left(x-k\\right)[\/latex].<\/li>\r\n\t
  2. Confirm that the remainder is 0.<\/li>\r\n\t
  3. Write the polynomial as the product of [latex]\\left(x-k\\right)[\/latex] and 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><\/div>\r\n
    \r\n
    \r\n
    \r\n

    Example 2: Using the Factor Theorem to Solve a Polynomial Equation<\/h3>\r\n

    Show that [latex]\\left(x+2\\right)[\/latex]\u00a0is a factor of [latex]{x}^{3}-6{x}^{2}-x+30[\/latex]. Find the remaining factors. Use the factors to determine the zeros of the polynomial<\/strong>.<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Solutions<\/h3>\r\n

    We can use synthetic division to show that [latex]\\left(x+2\\right)[\/latex] is a factor of the polynomial.<\/p>\r\n\r\n

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

    The remainder is zero, so [latex]\\left(x+2\\right)[\/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:<\/p>\r\n\r\n

    [latex]\\left(x+2\\right)\\left({x}^{2}-8x+15\\right)[\/latex]<\/div>\r\n

    We can factor the quadratic factor to write the polynomial as<\/p>\r\n\r\n

    [latex]\\left(x+2\\right)\\left(x - 3\\right)\\left(x - 5\\right)[\/latex]<\/div>\r\n

    By the Factor Theorem, the zeros of [latex]{x}^{3}-6{x}^{2}-x+30[\/latex] are \u20132, 3, and 5.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

    \r\n

    Try It 2<\/h3>\r\n

    Use the Factor Theorem to find the zeros of [latex]f\\left(x\\right)={x}^{3}+4{x}^{2}-4x - 16[\/latex]\u00a0given that [latex]\\left(x - 2\\right)[\/latex]\u00a0is a factor of the polynomial.<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

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

    [latex]f\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)+r[\/latex].<\/div>\n

    If k<\/em>\u00a0is a zero, then the remainder r<\/em>\u00a0is [latex]f\\left(k\\right)=0[\/latex]\u00a0and [latex]f\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)+0[\/latex]\u00a0or [latex]f\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)[\/latex].<\/p>\n

    Notice, written in this form, x<\/em>\u00a0\u2013\u00a0k<\/em> is a factor of [latex]f\\left(x\\right)[\/latex]. We can conclude if k\u00a0<\/em>is a zero of [latex]f\\left(x\\right)[\/latex], then [latex]x-k[\/latex] is a factor of [latex]f\\left(x\\right)[\/latex].<\/p>\n

    Similarly, if [latex]x-k[\/latex]\u00a0is a factor of [latex]f\\left(x\\right)[\/latex],\u00a0then the remainder of the Division Algorithm [latex]f\\left(x\\right)=\\left(x-k\\right)q\\left(x\\right)+r[\/latex]\u00a0is 0. This tells us that k<\/em>\u00a0is a zero.<\/p>\n

    This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree n<\/em>\u00a0in the complex number system will have n<\/em>\u00a0zeros. We can use the Factor Theorem to completely factor a polynomial into the product of n<\/em>\u00a0factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.<\/p>\n

    \n

    A General Note: The Factor Theorem<\/h3>\n

    According to the Factor Theorem<\/strong>, k<\/em>\u00a0is a zero of [latex]f\\left(x\\right)[\/latex]\u00a0if and only if [latex]\\left(x-k\\right)[\/latex]\u00a0is a factor of [latex]f\\left(x\\right)[\/latex].<\/p>\n<\/div>\n

    \n

    How To: Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.
    \n<\/strong><\/h3>\n
      \n
    1. Use synthetic division to divide the polynomial by [latex]\\left(x-k\\right)[\/latex].<\/li>\n
    2. Confirm that the remainder is 0.<\/li>\n
    3. Write the polynomial as the product of [latex]\\left(x-k\\right)[\/latex] and 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<\/div>\n
      \n
      \n
      \n

      Example 2: Using the Factor Theorem to Solve a Polynomial Equation<\/h3>\n

      Show that [latex]\\left(x+2\\right)[\/latex]\u00a0is a factor of [latex]{x}^{3}-6{x}^{2}-x+30[\/latex]. Find the remaining factors. Use the factors to determine the zeros of the polynomial<\/strong>.<\/p>\n<\/div>\n

      \n

      Solutions<\/h3>\n

      We can use synthetic division to show that [latex]\\left(x+2\\right)[\/latex] is a factor of the polynomial.<\/p>\n

      \n

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

      The remainder is zero, so [latex]\\left(x+2\\right)[\/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:<\/p>\n

      [latex]\\left(x+2\\right)\\left({x}^{2}-8x+15\\right)[\/latex]<\/div>\n

      We can factor the quadratic factor to write the polynomial as<\/p>\n

      [latex]\\left(x+2\\right)\\left(x - 3\\right)\\left(x - 5\\right)[\/latex]<\/div>\n

      By the Factor Theorem, the zeros of [latex]{x}^{3}-6{x}^{2}-x+30[\/latex] are \u20132, 3, and 5.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Try It 2<\/h3>\n

      Use the Factor Theorem to find the zeros of [latex]f\\left(x\\right)={x}^{3}+4{x}^{2}-4x - 16[\/latex]\u00a0given that [latex]\\left(x - 2\\right)[\/latex]\u00a0is a factor of the polynomial.<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

      \n\t\t\t

      Candela Citations<\/h3>\n\t\t\t\t\t
      \n\t\t\t\t\t\t
      \n\t\t\t\t\t\t\t
      CC licensed content, Shared previously<\/div>