1. Warm Up Add or subtract. 1. 2. 3. 4. x ≠ 0 13 24 x ≠ –1, x ≠ 1 Simplify. Identify any x-values for which the expression is undefined. 13 15 11 12 3 8 – 7 15 2 5 + x – 1 x 2 – 1 1 x + 1 1 3 x6x6 4x94x9 12x 3 EQ: How do I perform operations on Rational Functions?

2. Adding and subtracting rational expressions is similar to adding and subtracting fractions. To add or subtract rational expressions with like denominators, add or subtract the numerators and use the same denominator. EQ: How do I perform operations on Rational Functions?

3. Add or subtract. Identify any x-values for which the expression is undefined. Example 1A: Adding and Subtracting Rational Expressions with Like Denominators The expression is undefined at x = –4 because this value makes x + 4 equal 0. x – 3 x + 4 + x – 2 x + 4 EQ: How do I perform operations on Rational Functions?

4. Add or subtract. Identify any x-values for which the expression is undefined. Example 1B: Adding and Subtracting Rational Expressions with Like Denominators There is no real value of x for which x 2 + 1 = 0; the expression is always defined. 3x – 4 x 2 + 1 – 6x + 1 x 2 + 1 EQ: How do I perform operations on Rational Functions?

5. Check It Out! Example 1a Add or subtract. Identify any x-values for which the expression is undefined. 6x + 5 x 2 – 3 + 3x – 1 x 2 – 3 The expression is undefined at x = ± because this value makes x 2 – 3 equal 0. EQ: How do I perform operations on Rational Functions?

6. Check It Out! Example 1b Add or subtract. Identify any x-values for which the expression is undefined. 3x 2 – 5 3x – 1 – 2x 2 – 3x – 2 3x – 1 Distribute the negative sign. The expression is undefined at x = because this value makes 3x – 1 equal 0. 1 3 EQ: How do I perform operations on Rational Functions?

7. To add rational expressions with unlike denominators, rewrite both expressions with the LCD. This process is similar to adding fractions. EQ: How do I perform operations on Rational Functions?

8. Add. Identify any x-values for which the expression is undefined. Example 3A: Adding Rational Expressions x – 3 x 2 + 3x – 4 + 2x2x x + 4 EQ: How do I perform operations on Rational Functions?

9. x – 3 + 2x(x – 1) (x + 4)(x – 1) Add the numerators. 2x 2 – x – 3 (x + 4)(x – 1) Simplify the numerator. Example 3A Continued Add. Identify any x-values for which the expression is undefined. Write the sum in factored or expanded form. 2x 2 – x – 3 (x + 4)(x – 1) 2x 2 – x – 3 x 2 + 3x – 4 or The expression is undefined at x = –4 and x = 1 because these values of x make the factors (x + 4) and (x – 1) equal 0. EQ: How do I perform operations on Rational Functions?

10. Add. Identify any x-values for which the expression is undefined. x x + 2 + –8 x 2 – 4 Example 3B: Adding Rational Expressions EQ: How do I perform operations on Rational Functions?

11. x 2 – 2x – 8 (x + 2)(x – 2) Write the numerator in standard form. Add. Identify any x-values for which the expression is undefined. Example 3B Continued (x + 2)(x – 4) (x + 2)(x – 2) Factor the numerator. x – 4 x – 2 Divide out common factors. The expression is undefined at x = –2 and x = 2 because these values of x make the factors (x + 2) and (x – 2) equal 0. EQ: How do I perform operations on Rational Functions?

12. Check It Out! Example 3a Add. Identify any x-values for which the expression is undefined. 3x 2x – 2 + 3x – 2 3x – 3 EQ: How do I perform operations on Rational Functions?

13. Check It Out! Example 3a Continued Add. Identify any x-values for which the expression is undefined. 15x – 4 6(x – 1) Simplify the numerator. The expression is undefined at x = 1 because this value of x make the factor (x – 1) equal 0. Add the numerators. 9x + 6x – 4 6(x – 1) EQ: How do I perform operations on Rational Functions?

14. Add. Identify any x-values for which the expression is undefined. 2x + 6 x 2 + 6x + 9 + x x + 3 Warm UP EQ: How do I perform operations on Rational Functions?

15. Check It Out! Example 3b Continued Add. Identify any x-values for which the expression is undefined. Write the numerator in standard form. (x + 3)(x + 2) (x + 3)(x + 3) Factor the numerator. x + 2 x + 3 Divide out common factors. The expression is undefined at x = –3 because this value of x make the factors (x + 3) and (x + 3) equal 0. x 2 + 5x + 6 (x + 3)(x + 3) EQ: How do I perform operations on Rational Functions?

16. Example 4: Subtracting Rational Expressions Subtract. Identify any x- values for which the expression is undefined. 2x 2 – 30 x 2 – 9 – x + 3 x + 5 EQ: How do I perform operations on Rational Functions?

17. Example 4 Continued Subtract. Identify any x- values for which the expression is undefined. 2x 2 – 30 x 2 – 9 – x + 3 x + 5 2x 2 – 30 – x 2 – 2x + 15 (x – 3)(x + 3) Distribute the negative sign. x 2 – 2x – 15 (x – 3)(x + 3) Write the numerator in standard form. (x + 3)(x – 5) (x – 3)(x + 3) Factor the numerator. x – 5 x – 3 Divide out common factors. The expression is undefined at x = 3 and x = –3 because these values of x make the factors (x + 3) and (x – 3) equal 0. EQ: How do I perform operations on Rational Functions?

18. Subtract. Identify any x- values for which the expression is undefined. 3x – 2 2x + 52x + 5 – 5x – 2 2 Check It Out! Example 4a EQ: How do I perform operations on Rational Functions?

19. Check It Out! Example 4a Continued Distribute the negative sign. The expression is undefined at x = – and x = because these values of x make the factors (2x + 5) and (5x – 2) equal 0. 15x 2 – 16x + 4 – 4x – 10 (2x + 5)(5x – 2) 5 2 2 5 Subtract. Identify any x- values for which the expression is undefined. 3x – 2 2x + 52x + 5 – 5x – 2 2 EQ: How do I perform operations on Rational Functions?

20. Check It Out! Example 4b Subtract. Identify any x- values for which the expression is undefined. 2x 2 + 64 x 2 – 64 – x + 8 x – 4 EQ: How do I perform operations on Rational Functions?

21. Check It Out! Example 4b Distribute the negative sign. x 2 + 12x + 32 (x – 8)(x + 8) Write the numerator in standard form. (x + 8)(x + 4) (x – 8)(x + 8) Factor the numerator. x + 4 x – 8 Divide out common factors. The expression is undefined at x = 8 and x = –8 because these values of x make the factors (x + 8) and (x – 8) equal 0. Subtract. Identify any x- values for which the expression is undefined. 2x 2 + 64 x 2 – 64 – x + 8 x – 4 2x 2 + 64 – x 2 + 12x – 32) (x – 8)(x + 8) EQ: How do I perform operations on Rational Functions?

22. Lesson Quiz: Part I 2. 3x 2 – 2x + 7 (x – 2)(x + 1) x ≠ –1, 2 x – 9 x – 4 x ≠ 4, –4 Add or subtract. Identify any x-values for which the expression is undefined. 1. 2x + 1 x – 2 + x – 3 x + 1 x x + 4 – x + 36 x 2 – 16