How do you complete the following division: #(x^4 + x^3 - 5x^2 + 26x - 21)/(x^2 + 3x - 4)#?

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Answer 1 · 2016-12-11T01:42:23

#x^2 + 2x + 5 + 13/(5(x + 4)) + 2/(5(x - 1))#

Divide the numerator by the denominator using long division.

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So, #(x^4 + x^3 - 5x^2 + 26x - 21)/(x^2 + 3x - 4) = x^2 + 2x + 5 + (3x - 1)/(x^2 + 3x - 4)#.

We can now start the actual partial fraction decomposition process.

#x^2 + 3x - 4# can be factored as #(x +4)(x -1)#.

#A/(x + 4) + B/(x- 1) = (3x -1)/((x+ 4)(x - 1))#

#A(x - 1) + B(x +4) = 3x - 1#

#Ax - A + Bx + 4B = 3x - 1#

#(A + B)x + (4B - A) = 3x - 1#

We now write a system of equations:

#{(A + B = 3), (4B - A= -1):}#

Solve:

#B = 3 - A -> 4(3 - A) - A = -1#

#12 - 4A - A = -1#

#-5A = -13#

#A = 13/5#

#13/5 + B = 3#

#B = 2/5#

Therefore, the partial fraction decomposition of #(x^4 + x^3 - 5x^2 + 26x - 21)/(x^2 + 3x - 4)# is #x^2 + 2x + 5 + 13/(5(x + 4)) + 2/(5(x - 1))#.

Hopefully this helps!