Partial Fractions Lesson 8.5

Partial Fraction Decomposition Consider adding two algebraic fractions Partial fraction decomposition reverses the process

Partial Fraction Decomposition Motivation for this process The separate terms are easier to integrate

The Process Given Then f(x) can be decomposed with this cascading form Where polynomial P(x) has degree < n P(r) ≠ 0 Then f(x) can be decomposed with this cascading form

Strategy Given N(x)/D(x) If degree of N(x) greater than degree of D(x) divide the denominator into the numerator to obtain Degree of N1(x) will be less than that of D(x) Now proceed with following steps for N1(x)/D(x)

Strategy Factor the denominator into factors of the form where is irreducible For each factor the partial fraction must include the following sum of m fractions

Strategy Quadratic factors: For each factor of the form , the partial fraction decomposition must include the following sum of n fractions.

A Variation Suppose rational function has distinct linear factors Then we know

A Variation Now multiply through by the denominator to clear them from the equation Let x = 1 and x = -1 Solve for A and B

What If Single irreducible quadratic factor Then cascading form is But P(x) degree < 2m Then cascading form is

Gotta Try It Given Then

Gotta Try It Now equate corresponding coefficients on each side Solve for A, B, C, and D ?

Even More Exciting When but Example P(x) and D(x) are polynomials with no common factors D(x) ≠ 0 Example

Combine the Methods Consider where Express as cascading functions of P(x), D(x) have no common factors D(x) ≠ 0 Express as cascading functions of

Try It This Time Given Now manipulate the expression to determine A, B, and C

Partial Fractions for Integration Use these principles for the following integrals

Why Are We Doing This? Remember, the whole idea is to make the rational function easier to integrate

Assignment Lesson 8.5 Page 559 Exercises 1 – 29 EOO