How do you express 13x+2x3−1 in partial fractions?

1 Answer
Shwetank Mauria
Mar 24, 2016

Partial fractions are 5(x−1)+−5x+3x2+x+1

Explanation:

In 13x+2x3−1, we should first factorize x3−1, using identity (a3+b3)=(a+b)(a2−ab+b2).

Hence factors are given by (x3−1)=(x−1)(x2−x+1).

Partial fractions of 13x+2x3−1, will be

13x+2x3−1⇔Ax−1+Bx+Cx2+x+1and simplifying RHS it becomes

13x+2x3−1⇔A(x2+x+1)+(Bx+C)(x−1)(x−1)(x2+x+1) or

13x+2x3−1⇔(Ax2+Ax+A)+(Bx2−Bx+Cx−C)x3−1 or

13x+2x3−1⇔(A+B)x2+(A−B+C)x+(A−C)x3−1

Hence, we have A+B=0 - (i), A−B+C=13 - (ii) and A−C=2 - (iii).

From (i) we get B=−A, from (iii) C=A−2 and putting them in (ii),

we get A−(−A)+(A−2)=13 or 3A=15 or A=5

Hence B=−5 and C=3

Hence partial fractions are 5(x−1)+−5x+3x2+x+1

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