1. Chapter 4 Polynomials and Partial Fractions 4.1 Polynomials 4.3 Dividing Polynomials 4.5 Factor Theorem 4.2 Identities 4.4 Remainder Theorem 4.6 Solving Cubic Equations 4.7 Partial Fractions

2. Polynomials and Partial Fractions In this lesson, you will learn to decompose a rational expression into partial fractions. Rational functions with distinct linear factors, repeated linear factors and quadratic factors in the denominator are dealt with. 4.7 Partial Fractions Objectives

3. We will learn to decompose rational expressions into partial fractions. Polynomials and Partial Fractions If f(x) and g(x) are polynomials, then is called a rational expression. If the degree of the numerator f(x) is less than the degree of the denominator g(x), then is said to be proper, otherwise it is said to be improper.

4. Polynomials and Partial Fractions We will look at three categories of proper rational expressions and their corresponding partial fractions. Partial fraction is in the form Next, we will look at some examples. Denominator contains the factor Partial fraction is in the form

5. Substitute for A and B. Polynomials and Partial Fractions since x + 2 and x + 1 are both linear Common denominato r Equate numerators Make x + 1 zero. Make x + 2 zero. Example

6. Substitute for A, B and C. Polynomials and Partial Fractions x – 2 and x + 1 are linear and x + 1 is repeated. Common denominato r Equate numerators Make x + 1 zero. Make x – 2 zero. Make x = 0 and substitute for A and C. ( x + 1) is included just once in the denominator. Example

7. Substitute for A, B and C. Polynomials and Partial Fractions since x is linear and x 2 + 3 is quadratic Common denominato r Equate numerators Make x zero. Example

8. We will now learn to decompose improper rational expressions into partial fractions. Polynomials and Partial Fractions Before expressing improper fractions as partial fractions, we reduce the expression to the sum of a polynomial and a proper rational expression using long division. where Q(x) is the quotient and R(x) is the remainder If the degree of the numerator f(x) is greater than the degree of the denominator g(x), then is said to be improper.

9. Polynomials and Partial Fractions Subtract to find the remainder. Continue to find partial fractions. Divide 2x 3 by x 2. Subtract and bring down 33. Divide 5x 2 by x 2. Example

10. Substitute for A and B. Polynomials and Partial Fractions since x + 2 and x + 3 are both linear Common denominato r Equate numerators Make x + 3 zero. Make x + 2 zero.