Precalculus Examples

Split Using Partial Fraction Decomposition 4/(x(x-1))
Step 1
Decompose the fraction and multiply through by the common denominator.
Tap for more steps...
Step 1.1
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 .
Step 1.2
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 1.3
Cancel the common factor of .
Tap for more steps...
Step 1.3.1
Cancel the common factor.
Step 1.3.2
Rewrite the expression.
Step 1.4
Cancel the common factor of .
Tap for more steps...
Step 1.4.1
Cancel the common factor.
Step 1.4.2
Divide by .
Step 1.5
Simplify each term.
Tap for more steps...
Step 1.5.1
Cancel the common factor of .
Tap for more steps...
Step 1.5.1.1
Cancel the common factor.
Step 1.5.1.2
Divide by .
Step 1.5.2
Apply the distributive property.
Step 1.5.3
Move to the left of .
Step 1.5.4
Rewrite as .
Step 1.5.5
Cancel the common factor of .
Tap for more steps...
Step 1.5.5.1
Cancel the common factor.
Step 1.5.5.2
Divide by .
Step 1.6
Move .
Step 2
Create equations for the partial fraction variables and use them to set up a system of equations.
Tap for more steps...
Step 2.1
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.
Step 2.2
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.
Step 2.3
Set up the system of equations to find the coefficients of the partial fractions.
Step 3
Solve the system of equations.
Tap for more steps...
Step 3.1
Solve for in .
Tap for more steps...
Step 3.1.1
Rewrite the equation as .
Step 3.1.2
Divide each term in by and simplify.
Tap for more steps...
Step 3.1.2.1
Divide each term in by .
Step 3.1.2.2
Simplify the left side.
Tap for more steps...
Step 3.1.2.2.1
Dividing two negative values results in a positive value.
Step 3.1.2.2.2
Divide by .
Step 3.1.2.3
Simplify the right side.
Tap for more steps...
Step 3.1.2.3.1
Divide by .
Step 3.2
Replace all occurrences of with in each equation.
Tap for more steps...
Step 3.2.1
Replace all occurrences of in with .
Step 3.2.2
Simplify the right side.
Tap for more steps...
Step 3.2.2.1
Remove parentheses.
Step 3.3
Solve for in .
Tap for more steps...
Step 3.3.1
Rewrite the equation as .
Step 3.3.2
Add to both sides of the equation.
Step 3.4
Solve the system of equations.
Step 3.5
List all of the solutions.
Step 4
Replace each of the partial fraction coefficients in with the values found for and .