{"id":1939,"date":"2024-06-24T23:37:01","date_gmt":"2024-06-24T23:37:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1939"},"modified":"2025-08-15T02:46:57","modified_gmt":"2025-08-15T02:46:57","slug":"dividing-polynomials-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/dividing-polynomials-learn-it-2\/","title":{"raw":"Dividing Polynomials: Learn It 2","rendered":"Dividing Polynomials: Learn It 2"},"content":{"raw":"

The Division Algorithm<\/h2>\r\n
\r\n

the division algorithm<\/h3>\r\nThe Division Algorithm<\/strong> states that given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that\r\n\r\n \r\n

[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]<\/p>\r\n \r\n\r\n[latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].\r\n\r\n \r\n\r\nIf [latex]r\\left(x\\right)=0[\/latex],\u00a0then [latex]d\\left(x\\right)[\/latex]\u00a0divides evenly into [latex]f\\left(x\\right)[\/latex].\u00a0This means that both [latex]d\\left(x\\right)[\/latex]\u00a0and [latex]q\\left(x\\right)[\/latex]\u00a0are factors of [latex]f\\left(x\\right)[\/latex].\r\n\r\n<\/section>

How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial<\/strong>\r\n
    \r\n \t
  1. Set up the division problem.<\/li>\r\n \t
  2. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\r\n \t
  3. Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\r\n \t
  4. Subtract the bottom binomial from the terms above it.<\/li>\r\n \t
  5. Bring down the next term of the dividend.<\/li>\r\n \t
  6. Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\r\n \t
  7. If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\r\n<\/ol>\r\n<\/section>
    Divide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].[reveal-answer q=\"850001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"850001\"]
    \r\n\r\n[caption id=\"attachment_997\" align=\"aligncenter\" width=\"716\"]\"6x Steps of a division problem[\/caption]\r\n\r\n<\/center>There is a remainder of [latex]1[\/latex]. We can express the result as:\r\n

    [latex]\\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\frac{1}{3x - 2}[\/latex]<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nWe can check our work by using the Division Algorithm to rewrite the solution then multiplying.\r\n

    [latex]\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/p>\r\nNotice, as we write our result,\r\n

      \r\n \t
    • the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\r\n \t
    • the divisor is [latex]3x - 2[\/latex]<\/li>\r\n \t
    • the quotient is [latex]2{x}^{2}+5x - 7[\/latex]<\/li>\r\n \t
    • the remainder is [latex]1[\/latex]<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>
      [ohm2_question hide_question_numbers=1]24621[\/ohm2_question]<\/section>
      [ohm2_question hide_question_numbers=1]13812[\/ohm2_question]<\/section>","rendered":"

      The Division Algorithm<\/h2>\n
      \n

      the division algorithm<\/h3>\n

      The Division Algorithm<\/strong> states that given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that<\/p>\n

       <\/p>\n

      [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]<\/p>\n

       <\/p>\n

      [latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/p>\n

       <\/p>\n

      If [latex]r\\left(x\\right)=0[\/latex],\u00a0then [latex]d\\left(x\\right)[\/latex]\u00a0divides evenly into [latex]f\\left(x\\right)[\/latex].\u00a0This means that both [latex]d\\left(x\\right)[\/latex]\u00a0and [latex]q\\left(x\\right)[\/latex]\u00a0are factors of [latex]f\\left(x\\right)[\/latex].<\/p>\n<\/section>\n

      How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial<\/strong><\/p>\n
        \n
      1. Set up the division problem.<\/li>\n
      2. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\n
      3. Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\n
      4. Subtract the bottom binomial from the terms above it.<\/li>\n
      5. Bring down the next term of the dividend.<\/li>\n
      6. Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\n
      7. If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\n<\/ol>\n<\/section>\n
        Divide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].<\/p>\n