How do you write the partial fraction decomposition of the rational expression #(x^3 - 5x + 2) / (x^2 - 8x + 15)#?

Question

Elias Ackerman · Accepted Answer

#(x^3 - 5x + 3)/(x² - 8x + 15) = x + 8 + 45/2(1/(x - 3)) + 43/2(1/(x - 5))#

We need to do the division first. I am going to use long division, because I prefer it over synthetic:

.............................#x + 8# ............................__ #x² - 8x + 15)x^3 + 0x^2 - 5x + 3# ........................#-x^3 + 8x² -15x# .......................................#8x²-20x + 3# ...................................#-8x² + 64x - 120# .....................................................#44x - 117#

Check:

#(x + 8)(x² - 8x + 15) + 44x - 117 = #

#x³ - 8x² + 15x + 8x² -64x + 120 + 44x - 117 =#

#x³ - 5x + 3# This checks

#(x^3 - 5x + 3)/(x² - 8x + 15) = x + 8 + (44x - 177)/(x² - 8x + 15)#

Now we do the decomposition on the remainder:

#(44x - 177)/(x² - 8x + 15) = A/(x - 3) + B/(x - 5)#

#44x - 177 = A(x - 5) + B(x - 3)#

Let x = 3:

#44(3) - 177 = A(3 - 5) + B(3 - 3)#

#-45 = -2A#

#A = 45/2#

Let x = 5:

#44(5) - 177 = A(5 - 5) + B(5 - 3)#

#43 = 2B#

#B = 43/2#

#(x^3 - 5x + 3)/(x² - 8x + 15) = x + 8 + 45/2(1/(x - 3)) + 43/2(1/(x - 5))#