Question

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

Accepted Answer

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. 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. 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. 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: $$ 4x^2 + 5x - 9 = A(ax^2 + bx + c) + (Bx + C)(x - r) . $$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.

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

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Key Concepts

Polynomial Factorization

Partial Fraction Decomposition Setup

Solving Systems of Equations