Section 8.3 Partial Fraction Decomposition
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1 Section 8.6 Lecture Notes Page 1 of 10 Section 8.3 Partial Fraction Decomposition Partial fraction decomposition involves decomposing a rational function, or reversing the process of combining two or more rational expressions. The goal: end up with a sum of simpler rational functions (why?). 1 1 Example: x x Already know how to do this: What we re doing in this section is starting with the single rational expression, and breaking it down into parts: Looking at rational expressions of the form: Can only do this procedure for degree of the numerator is rational expressions where the the degree of the denominator. IF degree of numerator is degree of denominator (i.e. ) then start by doing long division:
2 Section 8.6 Lecture Notes Page of 10 The entire process depends on: Factors of the The first step is always to fully factor the denominator. When Q( is fully factored, it will consist of: Then determine how many different factors Q( has, and what type they are. There are four cases of types of factors: Case Description Example 1. Distinct Linear Factors Each linear factor only appears once in Q(. Repeated Linear Factors A linear factor appears more than once in Q( 3. Distinct Irreducible Quadratic Factors Each quadratic factor appears once in Q( 4. Repeated Irreducible Quadratic Factors A quadratic factor appears more than once IMPORTANT: each separate factor in Q( will account for a single in the partial fraction decomposition: A non-repeating factor counts as. A repeating factor counts for. The first step is to build the sum of terms.
3 Section 8.6 Lecture Notes Page 3 of 10 Example: P( Q( 3x ( x )( x 1) Example: P( Q( = x 1 (x+) (x 3)
4 Section 8.6 Lecture Notes Page 4 of 10 Example: P( Q( = 1 (x+1)(x +4) 3 P( x 1 Example: Q( ( x 16)
5 Section 8.6 Lecture Notes Page 5 of 10 After setting up the problem by building the sum of terms, the next step is to: Solve for the. The first step in solving is always: Get rid of the. Do this by multiplying the entire equation by. Example: Find the partial-fraction decomposition of the rational expression: 3x x +x Factor the denominator: Build the sum of terms: Multiply equation by Q(: Method 1 solve by Equating Coefficients: Two polynomials are equal to each other if and only if the coefficients of terms of like degree are. In other words, if: 4x 3 + 3x 5 = ax 3 + bx + cx + d then: a = b = c = d =
6 Section 8.6 Lecture Notes Page 6 of 10 Continue solving by equating coefficients: Multiply out stuff on right-hand side: 3x = A(x 1) + B(x + ) Group according to terms, and put into descending order: Factor out the x terms: Equate the coefficients: System of two equations in two variables, which can be solved by: 1.. Replace variables with their values to get final answer: Method solve by selecting specific values for x: Choose values of x, and substitute into both sides of the equation, in order to eliminate all but one of the variables. 3x = A(x 1) + B(x + )
7 Section 8.6 Lecture Notes Page 7 of 10 Example: Find the partial-fraction decomposition of the rational expression: 3 x 1 ( x 16) Factor the denominator: Build the sum of terms: Multiply equation by Q(: Use Method 1, and solve by equating coefficients: Multiply out stuff on right-hand side: Group according to terms, and put into descending order: Factor out the x terms: Equate the coefficients: Replace variables with their values to get final answer:
8 Section 8.6 Lecture Notes Page 8 of 10 Summary of Partial Fraction Decomposition Note: assumes that P( is already a proper rational expression. Q( 1. Fully the denominator, Q(. Determine the and of factors that Q has: 1. non-repeated linear factor. repeated linear factors 3. non-repeated quadratic factor 4. repeated quadratic factors 3. Build the of terms, accounting for each factor of Q as follows: Non-repeated linear factor one term = A1 x a Repeated linear factor multiple terms = A 1 x a + A (x a) + Ax+b Non-repeated quadratic factor one term = ax +bx+c repeated quadratic factor A1 x B1 A x B multiple terms =... ax bx c ( ax bx c) P( 4. Now have: sum of terms Q( Get rid of all denominators by multiplying through entire equation by 5. Solve for coefficients A, B, C, D, using the following methods, or using a combination of these methods: 1. By selecting specific values for x which will all coefficients except for one.. By coefficients. 6. Write the final answer.
9 Section 8.6 Lecture Notes Page 9 of 10 Find the partial-fraction decomposition of each rational expression. x x x 1. x (x +)
10 Section 8.6 Lecture Notes Page 10 of 10 Find the partial-fraction decomposition of each rational expression. 3. x+5 x 3 x +x 4. x 3 x x x x 1 Hint: factor the denominator by grouping.
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