1. Copyright © Cengage Learning. All rights reserved. Rational Expressions and Equations; Ratio and Proportion 6

2. Copyright © Cengage Learning. All rights reserved. Section 6.1 Simplifying Rational Expressions

3. 3 Objectives Identify all values of a variable for which a rational expression is undefined. Find the domain of a rational function. Write a rational expression in simplest form. Simplify a rational expression containing factors that are negatives. 1 1 2 2 3 3 4 4

4. 4 Simplifying Rational Expressions Fractions such as and that are the quotient of two integers are rational numbers. Expressions such as and where the denominators and/or numerators are polynomials, are called rational expressions. Since rational expressions indicate division, we must exclude any values of the variable that will make the denominator equal to 0.

5. 5 Simplifying Rational Expressions For example, a cannot be –2 in the rational expression because the denominator will be 0: When the denominator of a rational expression is 0, we say that the expression is undefined.

6. 6 Identify all values of a variable for which a rational expression is undefined. 1.

7. 7 Example Identify all values of x such that the following rational expression is undefined. Solution: To find the values of x that make the rational expression undefined, we set its denominator equal to 0 and solve for x. x 2 + x – 12 = 0 (x + 4)(x – 3) = 0 Factor the trinomial.

8. 8 Example – Solution x + 4 = 0 or x – 3 = 0 x = –4 x = 3 We can check by substituting 3 and –4 for x and verifying that these values make the denominator of the rational expression equal to 0. For x = 3 Apply the zero-factor property. Solve each equation. cont’d

9. 9 Example – Solution For x = –4 cont’d

10. 10 Example – Solution Since the denominator is 0 when x = 3 or x = –4, the rational expression is undefined at these values. cont’d

11. 11 Find the domain of a rational function 2.

12. 12 Find the domain of a rational function The same process is used to find the domain of a rational function. We know that the domain is the set of all values that can be substituted for the variable. We learned that the domain for the linear function f (x) = x + 3 was all real numbers.

13. 13 Example Find the domain for the rational function: Solution: We follow the steps in Example 1 to find the values that would make the expression undefined. Those values are 3 and –4. To write the domain, we must identify all values that can be substituted.

14. 14 Example – Solution Therefore, in set-builder notation, the domain of the rational function is {x | x  R, x  3, –4). This is read as “the set of all values of x such that x is a real number where x  3 and x  –4.” cont’d

15. 15 Write a rational expression in simplest form 3.

16. 16 Write a rational expression in simplest form We have seen that a fraction can be simplified by dividing out common factors shared by its numerator and denominator. For example, These examples illustrate the fundamental property of fractions.

17. 17 Write a rational expression in simplest form The Fundamental Property of Fractions If a, b, and x are real numbers, then Since rational expressions are fractions, we can use the fundamental property of fractions to simplify rational expressions. We factor the numerator and denominator of the rational expression and divide out all common factors.

18. 18 Write a rational expression in simplest form When all common factors have been divided out, we say that the rational expression has been written in simplest form.

19. 19 Example Simplify:. Assume that the denominator is not 0. Solution: We will factor the numerator and the denominator and then divide out any common factors, if possible. Factor the numerator and denominator. Divide out the common factors of 7, x, and y.

20. 20 Example – Solution This rational expression also can be simplified by using the rules of exponents. 2 – 1 = 1; 1 – 2 = –1 Multiply. cont’d

21. 21 Write a rational expression in simplest form In general, for any real number a, the following is true. Division by 1

22. 22 Simplify a rational expression containing factors that are negatives 4.

23. 23 Simplify a rational expression containing factors that are negatives If the terms of two polynomials are the same, except for signs, the polynomials are called negatives or opposites of each other. For example, x – y and y – x are negatives (opposites), 2a – 1 and 1 – 2a are negatives (opposites), and 3x 2 – 2x + 5 and –3x 2 + 2x – 5 are negatives (opposites). Example 10 shows why the quotient of two polynomials that are negatives is always –1.

24. 24 Example Simplify:. Assume that no denominators are 0. Solution: We can rearrange terms in each numerator, factor out –1, and proceed as follows:

25. 25 Example – Solution cont’d

26. 26 Simplify a rational expression containing factors that are negatives The previous example suggests this important result. Division of Negatives The quotient of any nonzero expression and its negative is –1. In symbols, we have If a  b, then.