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Chapter 11: Q. 18 (page 765)

Write the partial fraction decomposition of each rational expression.

3x(x+2)(x-4).

Short Answer

Expert verified

The partial fraction decomposition is1x+2+2x-4.

Step by step solution

01

Step 1. Given Information.

The rational expression is,

3x(x+2)(x-4).

02

Step 2. Partial Fraction Decomposition.

The decomposition of the partial fraction, we get,

3x(x+2)(x-4)=Ax+2+Bx-4............(1).

Multiplying both sides by (x+2)(x-4),

3x=A(x-4)+B(x+2)3x=(A+B)x+(-4A+2B).........(2)

Equating the coefficients of like powers of x, we get,

A+B=3........(3)-4A+2B=0鈬�-2A+B=0........(4)

Solving the equations we get,

A=1B=2

The partial fraction decomposition is,

3x(x+2)(x-4)=1x+2+2x-4.

03

Step 3. Checking the solution.

Adding the rational expressions,

1x+2+2x-4=1(x-4)+2(x+2)(x+2)(x-4)

=x-4+2x+4(x+2)(x-4)=3x(x+2)(x-4)

The solution is true.

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