Write the partial fraction decomposition of the given rational expression. 1x3−1=? What is the partial fraction decomposition?

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Write the partial fraction decomposition of the given rational expression.  1x3−1=?  What is the partial fraction decomposition?

Coon2000 · Accepted Answer

Step 1  The given rational expression is 1x3−1=?  Note that x3−1=(x−1)(x2+x+1)  Step 2  Let 1x3−1=Ax−1+Bx+Cx2+x+1  Remove the fraction and obtain that 1=A(x2+x+1)+(Bx+C)(x−1).  Substitute x=1 in 1=A(x2+x+1)+(Bx+C)(x−1),  1=A(12+1+1)+(B+C)(1−1)  1=3A+0  A=13  Substitute x=0 ∈ 1=A(x2+x+1)+(Bx+C)(x−1),  1=A(0+0+1)+C(0−1)  1=A−C  1=13−C  C=13−1  C=−23  Compare the coefficient of x2 ∈ 1=A(x2+x+1)+(Bx+C)(x−1) and obtain that,  0=A+B  B=−A  B=−13  Back substitute the obtained values, 1x3−1=13(x−1)+−13x−23x2+x+1  Thus, the required partial fraction is 1x3−1=13(x−1)−(x+2)3(x2+x+1).

Write the partial fraction decomposition of the given rational express

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