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Chapter 11: Q. 17 (page 765)

Write the partial fraction decomposition of each rational expression:

x(x-1)(x-2).

Short Answer

Expert verified

The partial fraction decomposition is,

x(x-1)(x-2)=-1x-1+2x-2.

Step by step solution

01

Step 1. Given Information

The rational expression is,

x(x-1)(x-2).
02

Step 2. Decompose the partial fractions.

Decompose the rational expressions,

x(x-1)(x-2)=Ax-1+Bx-2......(1)

Multiplying both sides by,

x=A(x-2)+B(x-1)x=(A+B)x+(-2A-B)...........(2)

Equating the coefficients of like powers of xto get,

A+B=1-2A-B=0

Solving the equations we get,

A=-1B=2

The partial fraction decomposition is,

x(x-1)(x-2)=-1x-1+2x-2.

03

Step 3. Checking the solution.

Adding the rational expressions,

-1x-1+2x-2=-1(x-2)+2(x-1)(x-1)(x-2)

=-x+2+2x-2(x-1)(x-2)=x(x-1)(x-2)

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