1. § 6.3 Complex Rational Expressions

2. Blitzer, Algebra for College Students, 6e – Slide #2 Section 6.3 Simplifying Complex Fractions Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more fractions. Woe is me, for I am complex. I am currently suffering from… a feeling of complexity

3. Blitzer, Algebra for College Students, 6e – Slide #3 Section 6.3 Complex Rational Expressions Simplifying a Complex Rational Expression by Multiplying by One in the Form T 1) Find the LCD of all rational expressions within the complex rational expression. 2) Multiply both the numerator and the denominator of the complex rational expression by this LCD. 3) Use the distributive property and multiply each term in the numerator and denominator by this LCD. Simplify. No fractional expressions should remain. 4) If possible, factor and simplify.

4. Blitzer, Algebra for College Students, 6e – Slide #4 Section 6.3 Simplifying Complex FractionsEXAMPLE Simplify: SOLUTION The denominators in the complex rational expression are 5 and x. The LCD is 5x. Multiply both the numerator and the denominator of the complex rational expression by 5x. Multiply the numerator and denominator by 5x.

5. Blitzer, Algebra for College Students, 6e – Slide #5 Section 6.3 Simplifying Complex Fractions Use the distributive property. CONTINUED Divide out common factors. Simplify. Factor and simplify.

6. Blitzer, Algebra for College Students, 6e – Slide #6 Section 6.3 Simplifying Complex Fractions Simplify. CONTINUED

7. Blitzer, Algebra for College Students, 6e – Slide #7 Section 6.3 Simplifying Complex FractionsEXAMPLE Simplify: SOLUTION The denominators in the complex rational expression are x + 6 and x. The LCD is (x + 6)x. Multiply both the numerator and the denominator of the complex rational expression by (x + 6)x. Multiply the numerator and denominator by (x + 6)x.

8. Blitzer, Algebra for College Students, 6e – Slide #8 Section 6.3 Simplifying Complex Fractions Use the distributive property. CONTINUED Divide out common factors. Simplify.

9. Blitzer, Algebra for College Students, 6e – Slide #9 Section 6.3 Simplifying Complex FractionsCONTINUED Subtract. Factor and simplify. Simplify.

10. Blitzer, Algebra for College Students, 6e – Slide #10 Section 6.3 Simplifying Complex Fractions Simplifying a Complex Rational Expression by Dividing 1) If necessary, add or subtract to get a single rational expression in the numerator. 2) If necessary, add or subtract to get a single rational expression in the denominator. 3) Perform the division indicated by the main fraction bar: Invert the denominator of the complex rational expression and multiply. 4) If possible, simplify.

11. Blitzer, Algebra for College Students, 6e – Slide #11 Section 6.3 Simplifying Complex FractionsEXAMPLE Simplify: SOLUTION 1) Subtract to get a single rational expression in the numerator.

12. Blitzer, Algebra for College Students, 6e – Slide #12 Section 6.3 Simplifying Complex Fractions 2) Add to get a single rational expression in the denominator. CONTINUED

13. Blitzer, Algebra for College Students, 6e – Slide #13 Section 6.3 Simplifying Complex Fractions 3) & 4) Perform the division indicated by the main fraction bar: Invert and multiply. If possible, simplify. CONTINUED

14. Blitzer, Algebra for College Students, 6e – Slide #14 Section 6.3 Simplifying Complex Fractions Important to Remember: Complex rational expressions have numerators or denominators containing one or more fractions. Complex rational expressions can be simplified by one of two methods presented in this section: (a) multiplying the numerator and denominator by the LCD (b) obtaining single expressions in the numerator and denominator and then dividing, using the definition of division for fractions