# Precalculus Examples

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 .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Rewrite the expression.

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Simplify each term.

Multiply by to get .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Apply the distributive property.

Move to the left of the expression .

Multiply by to get .

Multiply by to get .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Apply the distributive property.

Move to the left of the expression .

Multiply by to get .

Remove unnecessary parentheses.

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.

Replace all occurrences of with the solution found by solving the last equation for . In this case, the value substituted is .

Solve for in the second equation.

Replace all occurrences of with the solution found by solving the last equation for . In this case, the value substituted is .

Simplify the right side.

Multiply by to get .

Add and to get .

List the solutions to the system of equations.

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

Remove the zero from the expression.