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Chapter 5: Problem 17

Write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}+13 x-9}{x(x-1)(x+3)}$$

Short Answer

Expert verified
The partial fraction decomposition of the given rational expression is derived as a series of steps leading to an expression in the form: \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3}\) where A, B and C are obtained by solving equations or comparing coefficients.

Step by step solution

01

Set Up Expression with Undetermined Coefficients

Write the given rational expression \(\frac{4 x^{2}+13 x-9}{x(x-1)(x+3)}\) as a sum of simpler fractions with variables A, B, C as coefficients. This can be represented as: \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3}\).
02

Multiply through by denominator and Simplify

Multiply through by the common denominator \(x(x - 1)(x + 3)\) on both sides to remove the fractions. This leads to: \(4x^2 + 13x - 9 = A(x - 1)(x + 3) + Bx(x + 3) + Cx(x - 1)\). Then expand, simplify and collect terms.
03

Solve for A, B and C

To determine the coefficients A, B and C, select suitable values for x and solve the resulting equations. Also, it is possible to equate coefficients of like terms on both sides of the equation to solve for A, B, and C.
04

Substitute values of A, B and C to obtain the partial fraction decomposition

After finding values for A, B, and C, substitute them back into the expression obtained in Step 1 to get the partial fraction decomposition.

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