Find the partial fraction decomposition of 4x²+5x-9/(x³- 6x-9)

Verified step by step guidance

1. First, recognize that the expression is a rational function where the numerator is a polynomial of degree 2, and the denominator is a polynomial of degree 3: \( \frac{4x^2 + 5x - 9}{x^3 - 6x - 9} \). Since the degree of the numerator is less than the degree of the denominator, we can proceed directly to partial fraction decomposition without polynomial division.
2. Next, factor the denominator \( x^3 - 6x - 9 \). To do this, try to find at least one root by using the Rational Root Theorem or by inspection. Once a root \( r \) is found, factor out \( (x - r) \) using polynomial division or synthetic division to express the denominator as a product of a linear factor and a quadratic factor.
3. After factoring the denominator into \( (x - r)(ax^2 + bx + c) \), set up the partial fraction decomposition as \( \frac{4x^2 + 5x - 9}{(x - r)(ax^2 + bx + c)} = \frac{A}{x - r} + \frac{Bx + C}{ax^2 + bx + c} \), where \( A, B, \) and \( C \) are constants to be determined.
4. Multiply both sides of the equation by the denominator \( (x - r)(ax^2 + bx + c) \) to clear the fractions, resulting in an equation involving polynomials: \( 4x^2 + 5x - 9 = A(ax^2 + bx + c) + (Bx + C)(x - r) \).
5. Expand the right-hand side, collect like terms, and equate the coefficients of corresponding powers of \( x \) on both sides. This will give a system of equations in terms of \( A, B, \) and \( C \). Solve this system to find the values of these constants, completing the partial fraction decomposition.

Key Concepts

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler polynomials that multiply to give the original. For partial fraction decomposition, factoring the denominator into linear or irreducible quadratic factors is essential to set up the correct form of the decomposition.

Partial Fraction Decomposition Setup

This concept involves expressing a rational function as a sum of simpler fractions whose denominators are the factors of the original denominator. Each factor corresponds to a term with unknown coefficients, which are later solved to find the decomposition.

Solving Systems of Equations

After setting up the partial fractions, equate the numerators and solve the resulting system of linear equations to find the unknown coefficients. This step is crucial to complete the decomposition and express the original function as a sum of simpler fractions.