Express 2x/(x^3+5x^2+2x-8) as partial fractions.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The expression `(2x)/(x^3+5x^2+2x-8)` has to be expressed as partial fractions.

First determine the factors of x^3+5x^2+2x-8

x^3+5x^2+2x-8

=> x^3 + 4x^2 + x^2 + 4x - 2x - 8

=> x^2(x + 4) + x(x + 4) - 2(x + 4)

=> (x^2 + x - 2)(x + 4)

=> (x^2 + 2x - x - 2)(x + 4)

=> (x(x + 2) - 1(x + 2))(x + 4

=> (x - 1)(x + 2)(x + 4)

`(2x)/(x^3+5x^2+2x-8)`

=> `(2x)/((x - 1)(x + 2)(x + 4))`

=> `A/(x - 1) + B/(x + 2) + C/(x + 4)`

=> `(A(x^2 + 6x + 8) + B(x^2 + 3x - 4) + C(x^2 + x - 2))/((x - 1)(x + 2)(x + 4))`

=> `x^2(A + B + C) + x(6A + 3B + C) + 8A - 4B - 2C = 2x`

=> `A + B + C = 0, 6A + 3B + C = 2 and 8A - 4B - 2C = 0`

Solve for A, B and C

A + B + C = 0 ...(1)

6A + 3B + C = 2 ...(2)

4A - 2B - C = 0 ...(3)

(1) + (3)

=> 5A - B = 0

=> B = 5A

From (1), C = -A - B = -A - 5A

Substitute in (2)

=> 6A + 15A -6A = 2

=> 15A = 2

=> A = `2/15`

B = `2/3`

C = `-4/5`

The partial fraction expansion of ` ``(2x)/(x^3+5x^2+2x-8)` is `2/(15*(x - 1)) + 2/(3*(x+2)) - 4/(5*(x + 4))`

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