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Math Help - Partial Fractions

  1. #1
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    Partial Fractions

    Hi there
    Please could someone help me to solve this question?

    Express ((8x)^3 - (18x)^2) / (x-1)(4x+1)

    in the form Ax + B + (C/(x-1)) + (D/(4x+1))

    Thank you all for your help!
    Rachel
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  2. #2
    Super Member

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    Hello. Rachel!

    Express . \frac{(8x)^3 - (18x)^2}{(x-1)(4x+1)} . in the form: . Ax + B + \frac{C}{x-1} + \frac{D}{4x+1}
    Using long division, we have: . \frac{512x^3 - 324x^2}{4x^2 - 3x - 1} \;=\;128x + 15 + \frac{173x + 15}{(x-1)(4x+1)}


    Then: . \frac{173x + 15}{(x-1)(4x+1)} \;=\;\frac{C}{x-1} + \frac{D}{4x+1} \quad\Rightarrow\quad 173x + 15 \;=\;(4x+1)C + (x-1)D

    Let x = 1\!:\;\;188 \:=\:5C\quad\Rightarrow\quad C \:=\:\frac{188}{5}

    Let x = -\frac{1}{4}\!:\;\;-\frac{173}{4} + 15 \:=\:-\frac{5}{4}D \quad\Rightarrow\quad D \:=\:\frac{113}{5}


    Therefore: . \frac{512x^3 - 324x^2}{(x-1)(4x+1)} \;=\;128x + 15 + \frac{\frac{188}{5}}{x-1} + \frac{\frac{113}{5}}{4x+1}

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