How do you write the partial fraction decomposition of the rational expression # (x^3 - 6x^2 + 11x - 6) / (4x^3 - 28x^2 + 56x - 32) #?

1 Answer
George C.
Mar 21, 2016

#(x^3-6x^2+11x-6)/(4x^3-28x^2+56x-32)=1/4+1/(4(x-4))#

with exclusions #x != 1# and #x != 2#

Explanation:

Let us factor the numerator and denominator first:

#x^3-6x^2+11x-6#

#=(x-1)(x^2-5x+6)#

#=(x-1)(x-2)(x-3)#

#color(white)()#

#4x^3-28x^2+56x-32#

#=4(x^3-7x^2+14x-8)#

#=4(x-1)(x^2-6x+8)#

#=4(x-1)(x-2)(x-4)#

So:

#(x^3-6x^2+11x-6)/(4x^3-28x^2+56x-32)#

#=(color(red)(cancel(color(black)((x-1))))color(red)(cancel(color(black)((x-2))))(x-3))/(4color(red)(cancel(color(black)((x-1))))color(red)(cancel(color(black)((x-2))))(x-4))#

#=(x-3)/(4(x-4))#

#=(x-4+1)/(4(x-4))#

#=1/4+1/(4(x-4))#

with exclusions #x != 1# and #x != 2#

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