{"id":1212,"date":"2025-07-23T21:00:16","date_gmt":"2025-07-23T21:00:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1212"},"modified":"2026-03-20T21:36:18","modified_gmt":"2026-03-20T21:36:18","slug":"partial-fractions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/partial-fractions-learn-it-4\/","title":{"raw":"Partial Fractions: Learn It 4","rendered":"Partial Fractions: Learn It 4"},"content":{"raw":"

Repeated Irreducible Quadratic Factor<\/h2>\r\nNow that we can decompose a simplified rational expression with an irreducible quadratic factor, we will learn how to do partial fraction decomposition when the simplified rational expression has repeated<\/strong> irreducible quadratic factors. The decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented in increasing powers.\r\n\r\n
\r\n

partial fraction decomposition: repeated irreducible quadratic factor<\/h3>\r\nThe partial fraction decomposition of [latex]\\dfrac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex], when [latex]Q\\left(x\\right)[\/latex] has a repeated irreducible quadratic factor and the degree of [latex]P\\left(x\\right)[\/latex] is less than the degree of [latex]Q\\left(x\\right)[\/latex], is\r\n\r\n \r\n\r\n
[latex]\\dfrac{P\\left(x\\right)}{{\\left(a{x}^{2}+bx+c\\right)}^{n}}=\\dfrac{{A}_{1}x+{B}_{1}}{\\left(a{x}^{2}+bx+c\\right)}+\\dfrac{{A}_{2}x+{B}_{2}}{{\\left(a{x}^{2}+bx+c\\right)}^{2}}+\\dfrac{{A}_{3}x+{B}_{3}}{{\\left(a{x}^{2}+bx+c\\right)}^{3}}+\\cdot \\cdot \\cdot +\\dfrac{{A}_{n}x+{B}_{n}}{{\\left(a{x}^{2}+bx+c\\right)}^{n}}[\/latex]<\/center> \r\n\r\nWrite the denominators in increasing powers.\r\n\r\n<\/section>
How To: Given a rational expression that has a repeated irreducible factor, decompose it.<\/strong>\r\n
    \r\n \t
  1. Use variables like [latex]A,B[\/latex], or [latex]C[\/latex] for the constant numerators over linear factors, and linear expressions such as [latex]{A}_{1}x+{B}_{1},{A}_{2}x+{B}_{2}[\/latex], etc., for the numerators of each quadratic factor in the denominator written in increasing powers, such as\r\n
    [latex]\\dfrac{P\\left(x\\right)}{Q\\left(x\\right)}=\\dfrac{A}{ax+b}+\\dfrac{{A}_{1}x+{B}_{1}}{\\left(a{x}^{2}+bx+c\\right)}+\\dfrac{{A}_{2}x+{B}_{2}}{{\\left(a{x}^{2}+bx+c\\right)}^{2}}+\\cdots +\\text{ }\\dfrac{{A}_{n}+{B}_{n}}{{\\left(a{x}^{2}+bx+c\\right)}^{n}}[\/latex]<\/div><\/li>\r\n \t
  2. Multiply both sides of the equation by the common denominator to eliminate fractions.<\/li>\r\n \t
  3. Expand the right side of the equation and collect like terms.<\/li>\r\n \t
  4. Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.<\/li>\r\n<\/ol>\r\n<\/section>
    Decompose the given expression that has a repeated irreducible factor in the denominator.\r\n

    [latex]\\dfrac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\\left({x}^{2}+1\\right)}^{2}}[\/latex]<\/p>\r\n[reveal-answer q=\"901003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"901003\"]\r\n\r\nThe factors of the denominator are [latex]x,\\left({x}^{2}+1\\right)[\/latex], and [latex]{\\left({x}^{2}+1\\right)}^{2}[\/latex]. Recall that, when a factor in the denominator is a quadratic that includes at least two terms, the numerator must be of the linear form [latex]Ax+B[\/latex]. So, let\u2019s begin the decomposition.\r\n

    [latex]\\dfrac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\\left({x}^{2}+1\\right)}^{2}}=\\dfrac{A}{x}+\\dfrac{Bx+C}{\\left({x}^{2}+1\\right)}+\\dfrac{Dx+E}{{\\left({x}^{2}+1\\right)}^{2}}[\/latex]<\/p>\r\nWe eliminate the denominators by multiplying each term by [latex]x{\\left({x}^{2}+1\\right)}^{2}[\/latex]. Thus,\r\n

    [latex]{x}^{4}+{x}^{3}+{x}^{2}-x+1=A{\\left({x}^{2}+1\\right)}^{2}+\\left(Bx+C\\right)\\left(x\\right)\\left({x}^{2}+1\\right)+\\left(Dx+E\\right)\\left(x\\right)[\/latex]<\/p>\r\nExpand the right side.\r\n

    [latex]\\begin{align}&{x}^{4}+{x}^{3}+{x}^{2}-x+1=A\\left({x}^{4}+2{x}^{2}+1\\right)+B{x}^{4}+B{x}^{2}+C{x}^{3}+Cx+D{x}^{2}+Ex \\\\ &=A{x}^{4}+2A{x}^{2}+A+B{x}^{4}+B{x}^{2}+C{x}^{3}+Cx+D{x}^{2}+Ex \\end{align}[\/latex]<\/p>\r\nNow we will collect like terms.\r\n

    [latex]{x}^{4}+{x}^{3}+{x}^{2}-x+1=\\left(A+B\\right){x}^{4}+\\left(C\\right){x}^{3}+\\left(2A+B+D\\right){x}^{2}+\\left(C+E\\right)x+A[\/latex]<\/p>\r\nSet up the system of equations matching corresponding coefficients on each side of the equal sign.\r\n

    [latex]\\begin{align}A+B=1 \\\\ C=1 \\\\ \\\\ 2A+B+D=1 \\\\ C+E=-1 \\\\ A=1 \\end{align}[\/latex]<\/p>\r\nWe can use substitution from this point. Substitute [latex]A=1[\/latex] into the first equation.\r\n

    [latex]\\begin{align}1+B=1 \\\\ B=0 \\end{align}[\/latex]<\/p>\r\nSubstitute [latex]A=1[\/latex] and [latex]B=0[\/latex] into the third equation.\r\n

    [latex]\\begin{align}2\\left(1\\right)+0+D=1 \\\\ D=-1 \\end{align}[\/latex]<\/p>\r\nSubstitute [latex]C=1[\/latex] into the fourth equation.\r\n

    [latex]\\begin{align} 1+E=-1\\\\ E=-2\\end{align}[\/latex]<\/p>\r\nNow we have solved for all of the unknowns on the right side of the equal sign. We have [latex]A=1[\/latex], [latex]B=0[\/latex], [latex]C=1[\/latex], [latex]D=-1[\/latex], and [latex]E=-2[\/latex]. We can write the decomposition as follows:\r\n

    [latex]\\dfrac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\\left({x}^{2}+1\\right)}^{2}}=\\dfrac{1}{x}+\\dfrac{1}{\\left({x}^{2}+1\\right)}-\\dfrac{x+2}{{\\left({x}^{2}+1\\right)}^{2}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]321643[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]321644[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]321645[\/ohm_question]<\/section>","rendered":"

    Repeated Irreducible Quadratic Factor<\/h2>\n

    Now that we can decompose a simplified rational expression with an irreducible quadratic factor, we will learn how to do partial fraction decomposition when the simplified rational expression has repeated<\/strong> irreducible quadratic factors. The decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented in increasing powers.<\/p>\n

    \n

    partial fraction decomposition: repeated irreducible quadratic factor<\/h3>\n

    The partial fraction decomposition of [latex]\\dfrac{P\\left(x\\right)}{Q\\left(x\\right)}[\/latex], when [latex]Q\\left(x\\right)[\/latex] has a repeated irreducible quadratic factor and the degree of [latex]P\\left(x\\right)[\/latex] is less than the degree of [latex]Q\\left(x\\right)[\/latex], is<\/p>\n

     <\/p>\n

    [latex]\\dfrac{P\\left(x\\right)}{{\\left(a{x}^{2}+bx+c\\right)}^{n}}=\\dfrac{{A}_{1}x+{B}_{1}}{\\left(a{x}^{2}+bx+c\\right)}+\\dfrac{{A}_{2}x+{B}_{2}}{{\\left(a{x}^{2}+bx+c\\right)}^{2}}+\\dfrac{{A}_{3}x+{B}_{3}}{{\\left(a{x}^{2}+bx+c\\right)}^{3}}+\\cdot \\cdot \\cdot +\\dfrac{{A}_{n}x+{B}_{n}}{{\\left(a{x}^{2}+bx+c\\right)}^{n}}[\/latex]<\/div>\n

     <\/p>\n

    Write the denominators in increasing powers.<\/p>\n<\/section>\n

    How To: Given a rational expression that has a repeated irreducible factor, decompose it.<\/strong><\/p>\n
      \n
    1. Use variables like [latex]A,B[\/latex], or [latex]C[\/latex] for the constant numerators over linear factors, and linear expressions such as [latex]{A}_{1}x+{B}_{1},{A}_{2}x+{B}_{2}[\/latex], etc., for the numerators of each quadratic factor in the denominator written in increasing powers, such as\n
      [latex]\\dfrac{P\\left(x\\right)}{Q\\left(x\\right)}=\\dfrac{A}{ax+b}+\\dfrac{{A}_{1}x+{B}_{1}}{\\left(a{x}^{2}+bx+c\\right)}+\\dfrac{{A}_{2}x+{B}_{2}}{{\\left(a{x}^{2}+bx+c\\right)}^{2}}+\\cdots +\\text{ }\\dfrac{{A}_{n}+{B}_{n}}{{\\left(a{x}^{2}+bx+c\\right)}^{n}}[\/latex]<\/div>\n<\/li>\n
    2. Multiply both sides of the equation by the common denominator to eliminate fractions.<\/li>\n
    3. Expand the right side of the equation and collect like terms.<\/li>\n
    4. Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.<\/li>\n<\/ol>\n<\/section>\n
      Decompose the given expression that has a repeated irreducible factor in the denominator.<\/p>\n

      [latex]\\dfrac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\\left({x}^{2}+1\\right)}^{2}}[\/latex]<\/p>\n