1. Rational Expressions and Equations
Chapter 6
Rational Expressions and Equations

2. Complex Rational Expressions
6.5
Using Division to Simplify
Multiplying by the LCD

3. To Simplify a Complex Rational Expression by Dividing
1. Add or subtract, as needed, to get a single rational expression in the numerator.
2. Add or subtract, as needed, to get a single rational expression in the denominator.
3. Divide the numerator by the denominator (invert and multiply).
4. If possible, simplify by removing a factor equal to 1.

4. Simplify: Solution Rewriting with a division symbol
Multiplying by the reciprocal of the divisor (inverting and multiplying)
Factoring and removing a factor equal to 1.

5. Simplify:
Solution
Multiplying by 1 to get the LCD, 3, for the numerator.
Multiplying by 1 to get the LCD, 2x, for the denominator.
Adding
Subtracting

6. Rewriting with a division symbol. This is often done mentally.
Multiplying by the reciprocal of the divisor (inverting and multiplying)

7. To Simplify a Complex Rational Expression by Multiplying by the LCD
1. Find the LCD of all rational expressions within the complex rational expression.
2. Multiply the complex rational expression by a factor equal to 1. Write 1 as the LCD over itself (LCD/LCD).
3. Simplify. No fractional expressions should remain within the complex rational expression.
4. Factor and, if possible, simplify.

8. Solution Here we look for the LCD of all four factors.
Simplify:
Solution Here we look for the LCD of all four factors.
Multiplying by a factor equal to 1, using the LCD: 12/12=1
Multiplying the numerator by 12
Don’t forget the parentheses!
Multiplying the denominator by 12

9. Using the distributive law
Simplifying

10. Simplify: Solution The LCD is x.
When we multiply by x, all fractions in the numerator and denominator of the complex rational expression are cleared:
Using the distributive law

11. Simplify: Solution The LCD is x3 so we multiply by 1 using x3/x3.
Using the distributive law
All fractions have been cleared and simplified.