Write the partial fraction decomposition of each rational expression. 5x²+6x+3/(x + 1)(x² + 2x + 2)

Verified step by step guidance

1. Identify the form of the denominator. Here, the denominator is \( (x + 1)(x^2 + 2x + 2) \), which consists of a linear factor \( (x + 1) \) and an irreducible quadratic factor \( (x^2 + 2x + 2) \).
2. Set up the partial fraction decomposition with unknown constants. For the linear factor \( (x + 1) \), use a constant numerator \( A \). For the irreducible quadratic factor \( (x^2 + 2x + 2) \), use a linear numerator \( Bx + C \). So, write:
3. \[ \frac{5x^2 + 6x + 3}{(x + 1)(x^2 + 2x + 2)} = \frac{A}{x + 1} + \frac{Bx + C}{x^2 + 2x + 2} \]
4. Multiply both sides of the equation by the denominator \( (x + 1)(x^2 + 2x + 2) \) to clear the fractions:
5. \[ 5x^2 + 6x + 3 = A(x^2 + 2x + 2) + (Bx + C)(x + 1) \]
6. Expand the right-hand side and then collect like terms (powers of \( x \)) to form an equation where the coefficients of corresponding powers of \( x \) on both sides are equal. This will give a system of equations to solve for \( A \), \( B \), and \( C \).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions with denominators that are factors of the original denominator. This technique simplifies integration and other algebraic operations by breaking down complex expressions into manageable parts.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. Recognizing linear factors like (x + 1) and irreducible quadratic factors like (x² + 2x + 2) is essential for setting up the correct form of partial fractions.

Setting Up Partial Fractions for Linear and Quadratic Factors

When decomposing, linear factors correspond to terms with constants in the numerator (A/(x+1)), while irreducible quadratic factors require linear expressions in the numerator (Bx + C)/(x² + 2x + 2). Correctly assigning these forms is crucial for solving the decomposition.