Question

Write out the form of the partial fraction decomposition of the function (as in Example 6). Do not determine the numerical values of the coefficients.

(a)\(\frac{x}{{{x^2} + x - 2}}\)

Short answer

\(\frac{A}{{x - 1}} + \frac{B}{{x + 2}}\) is final answer.

Step-by-step solution

Given.

\(\frac{x}{{{x^2} + x - 2}}\)

Factorization of \({x^2} + x - 2 = \left( {x - 1} \right)\left( {x + 2} \right)\).

Calculation of partial fraction.

\(\frac{x}{{{x^2} + x - 2}} = \frac{x}{{\left( {x - 1} \right)\left( {x + 2} \right)}}\)

\(\frac{x}{{\left( {x - 1} \right)\left( {x + 2} \right)}} = \frac{A}{{x - 1}} + \frac{B}{{x + 2}}\)

Hence,\(\frac{A}{{x - 1}} + \frac{B}{{x + 2}}\)is a required answer.

(b)\(\frac{x}{{{x^2} + x + 2}}\)

Answer: (b)

\(1 - \frac{{\left( {x + 2} \right)}}{{{x^2} + x + 2}}\) is the final answer.

(b) Step 1: Given.

\(\frac{x}{{{x^2} + x + 2}}\)

(b) Step 2: Calculation of partial fraction.

\(\frac{x}{{{x^2} + x + 2}} = 1 - \frac{{\left( {x + 2} \right)}}{{{x^2} + x + 2}}\)

Further it can’t reduce.

Hence, \(1 - \frac{{\left( {x + 2} \right)}}{{{x^2} + x + 2}}\) is the final answer.