Problem 31 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{5 x^{2}+6 x+3}{(x+1)\left(x^{2}+2 x+2\right)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given expression \( \frac{5x^2 + 6x + 3}{(x+1)(x^2 + 2x + 2)} \) is obtained by finding constants A, B, and C from a set of linear equations. The resulting decomposition can then be written in the form \( \frac{A}{x + 1} + \frac{Bx + C}{x^2 + 2x + 2} \) .

Step 1: Set Up the Decomposition

The first step in Partial Fraction Decomposition is to set up the expression according to the factors in the denominator. For every linear factor \( (x + a) \), the form will be \( \frac{A}{x + a} \), and for every quadratic factor \( (x^2 + bx + c) \), the form will be \( \frac{Ax + B}{x^2 + bx + c} \). Applying these rules, the expression can be set up as: \[\frac{5x^2 + 6x + 3}{(x+1)(x^2 + 2x + 2)} = \frac{A}{x + 1} + \frac{Bx + C}{x^2 + 2x + 2}\]

Step 2: Equate Coefficients

The next step is to multiply through by the denominator on left side to get a polynomial equation. This will give:\(5x^2 + 6x + 3 = A(x^2 + 2x + 2) + (Bx + C)(x + 1)\)Now, by equating coefficients of like terms on both sides, we can establish three distinct equations to find the values of A, B, and C.

Step 3: Solve for Constants

Expanding and equating the coefficients, we get:For \(x^2\): \(5 = A + B\)For \(x\): \(6 = 2A + B + C\)And for constants: \(3 = 2A + C\)Solve these equations to find the values of A, B, and C.

Step 4: Substitute the Values back

After finding the values of A, B, and C, substitute them back into the partial fraction set up in Step 1 to achieve the final decomposition.