Denominators without Repeated Factors
A rational expression is an expression in the form , where and are polynomials and the value of is not zero. The degree of a term in a polynomial is the exponent of the variable. The degree of a polynomial is the degree of the term with the greatest degree. A linear factor of a polynomial is a factor in the form , where is a constant.
Sometimes it is helpful to rewrite a rational expression as a sum of simpler expressions. Partial fraction decomposition is a method of rewriting a rational expression as a sum of partial fractions, where each partial fraction is a rational expression with a denominator that has a degree less than the degree of .
To decompose a rational expression into a sum of partial fractions:
1. Factor the denominator of the rational expression.
2. Write an equation setting the original rational expression equal to a sum of partial fractions with unknown numerators. The numerator of each partial fraction must have a degree that is one less than the denominator.
3. Multiply both sides of the equation by the denominator of the original rational expression, which will eliminate the denominators.
4. Solve for the unknown values in the numerators.
5. Substitute the values for the numerators into the partial fractions.
Clear the denominators on each side of the equation in Step 2.Multiply both sides of the equation by . Then simplify the left side.
Denominators with Repeated Factors
Write the fraction as a sum with unknown numerators, using each linear factor once.If a factor is repeated times, the sum will include partial fractions with a constant numerator and the linear factor as the denominator. For each of those partial fractions, the exponent on the denominator will increase by 1. In this example, the factor occurs twice, so there are two partial fractions. The first has a denominator of , and the second has a denominator of .