Partial Fraction Decomposition

Partial fraction decomposition is a method for factoring fractions.

Description

The final factored form is determined by the factors of the denominator. For each factor, one or more new terms are introduced into the final form. Apply the rules from the following table with the understanding that A, B, C, and D are simply stand-ins for distinct numerator unknowns.

The steps of the method are:

1. Simplify the original fraction as much as possible.
2. Using the reduced denominator, formulate the factored final form with all numerator unknowns.
3. Multiply both sides by the denominator of the original fraction.

4. Try solving by using convenient values of x.

For example, if the problem has reduced to 3x + 4 = A(x + 2) + B(x + 3), try substituting in values that will cancel out entire terms, like x = -2 and x = -3.

This is not always possible, in which continue to:

5. Gather like terms and set up a linear system of equations.

For example, if the problem factors to x² - 5x + 9 = A(x² + 2)(x - 1) + B(x² + 2) + C(x - 1)², begin expanding and gathering terms by polynomial.

• A(x³ - x² + 2x - 2) + B(x² + 2) + C(x² - 2x + 1)

• Ax³ - Ax² + A2x - A2 + Bx² + B2 + Cx² - C2x + C

• x³(A) + x²(-A + B + C) + x(2A - 2C) + (-A2 + B2 + C)

Now taking into account the LHS, the linear system of equations is:

• x³(0) = x³(A) or 0 = A

• x²(1) = x²(-A + B + C) or 1 = -A + B + C

• x(-5) = x(2A - 2C) or -5 = 2A - 2C

• 9 = -A2 + B2 + C

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