# Precalculus Examples

Factor using the AC method.

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Write the factored form using these integers.

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place .

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place .

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of .

Cancel the common factor.

Divide by .

Simplify each term.

Cancel the common factor of .

Cancel the common factor.

Divide by .

Apply the distributive property.

Multiply by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Apply the distributive property.

Move to the left of .

Move .

Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

Set up the system of equations to find the coefficients of the partial fractions.

Solve for in the first equation.

Rewrite the equation as .

Subtract from both sides of the equation.

Replace all occurrences of in with .

Subtract from .

Solve for in the second equation.

Rewrite the equation as .

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Subtract from .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Replace all occurrences of in with .

Simplify .

Multiply by .

Subtract from .

Replace each of the partial fraction coefficients in with the values found for and .