1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Introduction to Rational Expressions Summary of Arithmetic Rules for Rational Numbers (or Fractions) A fraction (or rational number) is a number that can be written in the form where a and b are integers and b ≠ 0. (Remember, no denominator can be 0.) The Fundamental Principle: where b, k ≠ 0

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Introduction to Rational Expressions Summary of Arithmetic Rules for Rational Numbers (or Fractions) (cont.) The reciprocal of where a, b ≠ 0. Multiplication: where b, d ≠ 0 Division: where b, c, d ≠ 0

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Introduction to Rational Expressions Summary of Arithmetic Rules for Rational Numbers (or Fractions) (cont.) Addition: where b ≠ 0 Subtraction: where b ≠ 0

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: Developmental Mathematics Section 13.1: Multiplication and Division with Rational Expressions

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Introduction to Rational Expressions Rational Expressions A rational expression is an algebraic expression that can be written in the form where P and Q are polynomials and Q ≠ 0. Remember, the denominator of a rational expression can never be 0. Division by 0 is undefined.

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Finding Restrictions on the Variable

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Introduction to Rational Expressions The Fundamental Principle of Rational Expressions If is a rational expression and P, Q, and K are polynomials where Q, K ≠ 0, then

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Reducing Rational Expressions

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Reducing (or Simplifying) Rational Expressions Opposites in Rational Expressions For a polynomial P, where P ≠ 0. In particular, where x ≠ a.

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Reducing (or Simplifying) Rational Expressions Common Error “Divide out” only common factors. Wrong SolutionCorrect Solution

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Reducing (or Simplifying) Rational Expressions Common Error Wrong SolutionCorrect Solution

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Multiplication with Rational Expressions Multiply and reduce, if possible. Use the rules for exponents when they apply. State any restrictions on the variable(s).

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Multiplication with Rational Expressions Multiply and simplify:

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Multiplication with Rational Expressions Multiply and simplify:

15. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Division with Rational Expressions

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Division with Rational Expressions

17. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Division with Rational Expressions