{"id":303,"date":"2015-09-18T21:08:24","date_gmt":"2015-09-18T21:08:24","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=303"},"modified":"2015-11-03T22:20:38","modified_gmt":"2015-11-03T22:20:38","slug":"factoring-the-greatest-common-factor-of-a-polynomial","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/factoring-the-greatest-common-factor-of-a-polynomial\/","title":{"raw":"Factoring the Greatest Common Factor of a Polynomial","rendered":"Factoring the Greatest Common Factor of a Polynomial"},"content":{"raw":"When we study fractions, we learn that the greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex] and [latex]20[\/latex] The GCF of polynomials works the same way: [latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest polynomial that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].\r\n\r\nWhen factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.\r\n
\r\n

A General Note: Greatest Common Factor<\/h3>\r\nThe greatest common factor<\/strong> (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.\r\n\r\n<\/div>\r\n
\r\n

How To: Given a polynomial expression, factor out the greatest common factor.\r\n<\/strong><\/h3>\r\n
    \r\n\t
  1. Identify the GCF of the coefficients.<\/li>\r\n\t
  2. Identify the GCF of the variables.<\/li>\r\n\t
  3. Combine to find the GCF of the expression.<\/li>\r\n\t
  4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.<\/li>\r\n\t
  5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Example 1: Factoring the Greatest Common Factor<\/h3>\r\nFactor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nFirst, find the GCF of the expression. The GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex]. The GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex]. (Note that the GCF of a set of expressions in the form [latex]{x}^{n}[\/latex] will always be the exponent of lowest degree.) And the GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[\/latex].\r\n\r\nNext, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\\left(2{x}^{2}{y}^{2}\\right)=6{x}^{3}{y}^{3},3xy\\left(15xy\\right)=45{x}^{2}{y}^{2}[\/latex], and [latex]3xy\\left(7\\right)=21xy[\/latex].\r\n\r\nFinally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.\r\n
    [latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)[\/latex]<\/div>\r\n<\/div>\r\n
    \r\n

    Analysis of the Solution<\/h3>\r\nAfter factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Try It 1<\/h3>\r\nFactor [latex]x\\left({b}^{2}-a\\right)+6\\left({b}^{2}-a\\right)[\/latex] by pulling out the GCF.\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

    When we study fractions, we learn that the greatest common factor<\/strong> (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[\/latex] is the GCF of [latex]16[\/latex] and [latex]20[\/latex] because it is the largest number that divides evenly into both [latex]16[\/latex] and [latex]20[\/latex] The GCF of polynomials works the same way: [latex]4x[\/latex] is the GCF of [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex] because it is the largest polynomial that divides evenly into both [latex]16x[\/latex] and [latex]20{x}^{2}[\/latex].<\/p>\n

    When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.<\/p>\n

    \n

    A General Note: Greatest Common Factor<\/h3>\n

    The greatest common factor<\/strong> (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.<\/p>\n<\/div>\n

    \n

    How To: Given a polynomial expression, factor out the greatest common factor.
    \n<\/strong><\/h3>\n
      \n
    1. Identify the GCF of the coefficients.<\/li>\n
    2. Identify the GCF of the variables.<\/li>\n
    3. Combine to find the GCF of the expression.<\/li>\n
    4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.<\/li>\n
    5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.<\/li>\n<\/ol>\n<\/div>\n
      \n

      Example 1: Factoring the Greatest Common Factor<\/h3>\n

      Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      First, find the GCF of the expression. The GCF of [latex]6,45[\/latex], and [latex]21[\/latex] is [latex]3[\/latex]. The GCF of [latex]{x}^{3},{x}^{2}[\/latex], and [latex]x[\/latex] is [latex]x[\/latex]. (Note that the GCF of a set of expressions in the form [latex]{x}^{n}[\/latex] will always be the exponent of lowest degree.) And the GCF of [latex]{y}^{3},{y}^{2}[\/latex], and [latex]y[\/latex] is [latex]y[\/latex]. Combine these to find the GCF of the polynomial, [latex]3xy[\/latex].<\/p>\n

      Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]3xy\\left(2{x}^{2}{y}^{2}\\right)=6{x}^{3}{y}^{3},3xy\\left(15xy\\right)=45{x}^{2}{y}^{2}[\/latex], and [latex]3xy\\left(7\\right)=21xy[\/latex].<\/p>\n

      Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.<\/p>\n

      [latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)[\/latex]<\/div>\n<\/div>\n
      \n

      Analysis of the Solution<\/h3>\n

      After factoring, we can check our work by multiplying. Use the distributive property to confirm that [latex]\\left(3xy\\right)\\left(2{x}^{2}{y}^{2}+15xy+7\\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[\/latex].<\/p>\n<\/div>\n

      \n

      Try It 1<\/h3>\n

      Factor [latex]x\\left({b}^{2}-a\\right)+6\\left({b}^{2}-a\\right)[\/latex] by pulling out the GCF.<\/p>\n

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

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