Partial Fractions Lesson 8 5 Partial Fraction Decomposition

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 § 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) 1. If degree of N(x) greater than degree of D(x) divide the denominator into the numerator to obtain Degree of N 1(x) will be less than that of D(x) § Now proceed with following steps for N 1(x)/D(x)

Strategy 2. Factor the denominator into factors of the form where is irreducible 3. For each factor the partial fraction must include the following sum of m fractions

Strategy 4. 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 § But P(x) degree < 2 m • 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 § P(x) and D(x) are polynomials with no common factors § D(x) ≠ 0 • Example

Combine the Methods • Consider where § 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