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Resolve the fraction (2x² + 7x + 33) / (x³ + x² - 11x) into partial fractions.

Mathematics

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Step-by-step solution

10 steps

  1. Factor the denominator.

    x3+2x2−11x=x(x2+2x−11)x^3 + 2x^2 - 11x = x(x^2 + 2x - 11)

  2. Set up the partial fraction decomposition.

    2x2+7x+23x(x2+2x−11)=Ax+Bx+Cx2+2x−11\frac{2x^2 + 7x + 23}{x(x^2 + 2x - 11)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 2x - 11}

  3. Multiply through by the denominator x(x2+2x−11)x(x^2 + 2x - 11).

    2x2+7x+23=A(x2+2x−11)+(Bx+C)x2x^2 + 7x + 23 = A(x^2 + 2x - 11) + (Bx + C)x

  4. Expand the right-hand side.

    A(x2+2x−11)+(Bx+C)x=Ax2+2Ax−11A+Bx2+Cx=(A+B)x2+(2A+C)x−11AA(x^2 + 2x - 11) + (Bx + C)x = Ax^2 + 2Ax - 11A + Bx^2 + Cx = (A + B)x^2 + (2A + C)x - 11A

  5. Equate coefficients of corresponding powers of xx.

    {A+B=22A+C=7−11A=23\begin{cases} A + B = 2 \\ 2A + C = 7 \\ -11A = 23 \end{cases}

  6. Solve for AA.

    A=23−11=−2311A = \frac{23}{-11} = -\frac{23}{11}

  7. Solve for BB.

    B=2−A=2−(−2311)=2+2311=2211+2311=4511B = 2 - A = 2 - \left(-\frac{23}{11}\right) = 2 + \frac{23}{11} = \frac{22}{11} + \frac{23}{11} = \frac{45}{11}

  8. Solve for CC.

    C=7−2A=7−2(−2311)=7+4611=7711+4611=12311C = 7 - 2A = 7 - 2\left(-\frac{23}{11}\right) = 7 + \frac{46}{11} = \frac{77}{11} + \frac{46}{11} = \frac{123}{11}

  9. Substitute AA, BB, and CC into the decomposition.

    2x2+7x+23x3+2x2−11x=−2311x+4511x+12311x2+2x−11\frac{2x^2 + 7x + 23}{x^3 + 2x^2 - 11x} = \frac{-\frac{23}{11}}{x} + \frac{\frac{45}{11}x + \frac{123}{11}}{x^2 + 2x - 11}

  10. Simplify the expression.

    2x2+7x+23x3+2x2−11x=111(−23x+45x+123x2+2x−11)\frac{2x^2 + 7x + 23}{x^3 + 2x^2 - 11x} = \frac{1}{11}\left( \frac{-23}{x} + \frac{45x + 123}{x^2 + 2x - 11} \right) 111(−23x+45x+123x2+2x−11)\boxed{\dfrac{1}{11}\left( \dfrac{-23}{x} + \dfrac{45x + 123}{x^{2} + 2x - 11} \right)}
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