# Calculus Examples

Factor the fraction.

Remove parentheses.

Factor the denominator.

Remove parentheses.

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 .

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.

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 .

Simplify the expression.

Remove unnecessary parentheses.

Reorder and .

Reorder and .

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.

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.

The original partial fraction expression contains the variables from the system of equations.

Remove the zero from the expression.