A rational expression is the quotient of two polynomials. The domain of a rational expression excludes numbers that make the denominator equal to zero. The document provides examples and steps for simplifying, multiplying, dividing, adding, and subtracting rational expressions, including those with complex fractions.

1. Section P6 Rational Expressions

2. Rational Expressions

3. A rational expression is the quotient of two polynomials. The set of real numbers for which an algebraic expression is defined is the domain of the expression. Because division by zero is undefined, we must exclude numbers from a rational expression’s domain that make the denominator zero. See examples below.

4. Example What numbers must be excluded from the domain?

5. Simplifying Rational Expressions

7. Example Simplify and indicate what values are excluded from the domain:

8. Example Simplify and indicate what values are excluded from the domain:

9. Multiplying Rational Expressions

11. Example Multiply and Simplify:

12. Dividing Rational Expressions

13. We find the quotient of two rational expressions by inverting the divisor and multiplying.

14. Example Divide and Simplify:

15. Adding and Subtracting Rational Expressions with the Same Denominator

16. Add or subtract rational expressions with the same denominator by (1) Adding or subtracting the numerators, (2) Placing this result over the common denominator, and (3) Simplifying, if possible.

17. Example Add:

18. Example Subtract:

19. Adding and Subtracting Rational Expressions with Different Denominators

21. Example Subtract:

22. Example Add:

26. Example Add:

27. Example Add:

28. Complex Rational Expresisons

29. Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more rational expressions. Here are two examples of such expressions listed below:

30. Example Simplify:

31. Example Simplify:

32. (a) (b) (c) (d) Simplify:

33. (a) (b) (c) (d) Divide