Precalculus Examples

Split Using Partial Fraction Decomposition
Decompose the fraction and multiply through by the common denominator.
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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.
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Cancel the common factor.
Rewrite the expression.
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by .
Simplify each term.
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Multiply by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by .
Apply the distributive property.
Move to the left of the expression .
Multiply by .
Multiply by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by .
Apply the distributive property.
Move to the left of the expression .
Multiply by .
Remove unnecessary parentheses.
Create equations for the partial fraction variables and use them to set up a system of equations.
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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 the system of equations.
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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 .
Multiply by .
List the solutions to the system of equations.
Replace each of the partial fraction coefficients in with the values found for , and .
Move the negative in front of the fraction.
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