How do I find the partial-fraction decomposition of #(-3x^3 + 8x^2 - 4x + 5)/(-x^4 + 3x^3 - 3x^2 + 3x - 2)#?

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Answer 1 · 2014-09-14T19:24:38

Break the polynomial into two groups where each group is factorable on its own.

#(-3x^3+8x^2-4x+5)/((-x^4-3x^2-2)+(3x^3+3)#

Factor each group.

#(-3x^3+8x^2-4x+5)/(1(x^2+2)(x^2+1)+3x(x^2+1)#

Factor out the corresponding factor of #x^2+1# from each group.

#(-3x^3+8x^2-4x+5)/((-x^2-2)(x^2-1)+(3x)(x^2+1))#

Combine the individual factors into a single factored expression.

#(-3x^3+8x^2- 4x+5)/((x^2+2)(-x^2+3x-2))#

In this problem #-1*-2=2# and #-1-2=-3#, so insert -1 as the right hand term of one factor and -2 as the right-hand term of the other factor.

#(-3x^3+8x^2-4x+54)/(-(x^2+1)(x-1)(x+2)#

Multiply the numerator by -1.

#(3x^3-8x^2+4x-5)/((x^2+1)(x-1)(x-2))#