# Split Using Partial Fraction Decomposition (x^3+x^2-8)/((x^2+3)^2)

Decompose the fraction and multiply through by the common denominator.
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 is 2nd order, terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.
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 is 2nd order, terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.
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.
Divide by .
Simplify each term.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Multiply by .
Cancel the common factor.
Rewrite the expression.
Divide by .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify each term.
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Move to the left of .
Move to the left of .
Simplify the expression.
Move .
Reorder and .
Move .
Move .
Move .
Create equations for the partial fraction variables and use them to set up a system of equations.
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 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 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.
Rewrite the equation as .
Replace all occurrences of in with .
Multiply by .
Rewrite the equation as .
Replace all occurrences of in with .
Multiply by .
Solve for in the third equation.
Rewrite the equation as .
Subtract from both sides of the equation.
Solve for in the fourth 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 .
Replace each of the partial fraction coefficients in with the values found for , , , and .
Simplify.
Remove parentheses.
Multiply by .
Multiply by .
Split Using Partial Fraction Decomposition (x^3+x^2-8)/((x^2+3)^2)