Problem 17 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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: \(\frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 3}\).

Step 2: 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.

Step 3: 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.

Step 4: 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.