1. Partial Fractions MATH 109 - Precalculus S. Rook

2. Overview Section 7.4 in the textbook: – Decomposition with linear factors – Decomposition with quadratic factors – Decomposition with improper rational expressions 2

3. Partial Decomposition with Linear Factors

4. Decomposition in General Partial fraction decomposition: the process of breaking up a large fraction into smaller, more manageable (hopefully) fractions – We know how to combine multiple fractions into a single fraction: e.g. – Decomposition reverses the process e.g. Depends on the denominator – Attempt to factor the denominator as completely as possible 4

5. Unique Linear Factors A linear factor has the form px + q where p and q are constants Each unique linear factor in the denominator contributes a partial fraction of to the decomposition where A is a constant – Writing the partial fraction contributed by each factor in the denominator is called writing the general form of the decomposition – e.g. What would be the general form of the decomposition of 5

6. Writing the Basic Equation Once we have written the general form of the decomposition, we need to solve for the constants in the numerator The original fraction is equivalent to the sum of the partial fractions: Eliminate the fractions by multiplying both sides by the LCM: – This is called the basic equation and it holds true for any value of x 6

7. Solving the Basic Equation for Linear Factors For linear factors, pick convenient values for x in order find the value of each constant: – e.g. What value of x makes it easiest to find A? – e.g. What value of x makes it easiest to find B? – This process is straightforward when all factors of the denominator in the original fraction are linear We will discuss how to handle solving for the constants of the partial fractions for quadratic factors later After solving for the value of each constant, write the final form of the decomposition Can check whether decomposition matches the original fraction 7

8. Decomposition with Unique Linear Factors (Example) Ex 1: Write the partial fraction decomposition of the rational expression: a) b) 8

9. Repeated Linear Factors Each factor of the form (px + q) n in the factored denominator of the original fraction contributes the sum of partial fractions to the decomposition n > 1 where n is an integer; if n = 1, the factor is unique – e.g. What would be the general form of the decomposition of We can solve for the constants using the technique developed for unique linear factors 9

10. Decomposition with Repeated Linear Factors (Example) Ex 2: Write the partial fraction decomposition of the rational expression: a) b) 10

11. Decomposition with Quadratic Factors

12. Unique Quadratic Factors A quadratic factor has the form ax 2 + bx + c where the factor is irreducible and a, b, and c are constants – i.e. Cannot be factored further over the reals – e.g. (x 2 – 1) is NOT a quadratic factor since it can be reduced to (x + 1)(x – 1) – e.g. (x 2 + 1) IS a quadratic factor since it is irreducible Each unique quadratic factor in the denominator contributes a partial fraction of to the decomposition where B and C are constants – e.g. What would be the general form of the decomposition of 12

13. Solving the Basic Equation for Quadratic Factors When working with partial fractions contributed by quadratic factors, it is often difficult to find convenient values of x like we were able to with the linear factors Set up the basic equation: Expand and group by powers of x: Construct a system of equations by equating the coefficients of each power of x on the left side with its partner on the right side: 13

14. Repeated Quadratic Factors Each factor of the form (ax 2 + bx + c) n in the factored denominator of the original fraction contributes the sum of partial fractions to the decomposition n > 1 where n is an integer; if n = 1, the factor is unique – e.g. What would be the general form of the decomposition of We can solve for the constants using the technique developed for unique quadratic factors 14

15. Decomposition with Quadratic Factors (Example) Ex 3: Write the partial fraction decomposition of the rational expression: a) b) 15

16. Decomposition with Improper Rational Expressions

17. Improper Rational Expressions An improper rational expression occurs when the degree of its numerator is greater than OR equal to the degree of its denominator – e.g. or Thus far, we have worked exclusively with proper rational expressions – i.e. the degree of the numerator is less than the degree of the denominator 17

18. Decomposition with Improper Rational Expressions Before decomposing a rational expression, we MUST check to see whether it is proper – If the rational expression is improper, we MUST perform a long division Perform a decomposition on the remainder (now proper) Do not forget the quotient of the initial long division – If the rational expression is proper, proceed as normal i.e. factor the denominator and look for linear and quadratic factors The first step in a decomposition problem should ALWAYS be to check whether the rational expression is proper 18

19. Decomposition with Improper Rational Expressions (Example) Ex 4: Write the partial fraction decomposition of the rational expression: a) b) 19

20. Summary After studying these slides, you should be able to: – Decompose rational expressions Additional Practice – See the list of suggested problems for 7.4 Next lesson – Sequences & Series (Section 9.1) 20