1. Warm-Up Exercises Section 5.5 Adding and Subtracting Rational Expressions

2. Warm-Up Exercises HW Quiz

3. Warm-Up Exercises Add the following fractions

4. Warm-Up Exercises Complex Fractions

5. Warm-Up Exercises EXAMPLE 1 Add or subtract with like denominators Perform the indicated operation. 7 4x4x + 3 4x4x a. 2x2x x + 6 – 5 b. SOLUTION 7 4x4x + 3 4x4x a. = 7 + 3 4x4x 10 4x4x = 5 2x2x = Add numerators and simplify result. x + 6 2x – 5 = 2x2x x + 6 5 – b. Subtract numerators.

6. Warm-Up Exercises GUIDED PRACTICE for Example 1 Perform the indicated operation and simplify. 1. 7 12x − 5 2. 2 3x 2 + 1 3x23x2 3. 4x x–2 – x 4. 2x 2 x 2 +1 + 2 1 6x6x ANSWER 1 x 2 ANSWER 3x x – 2 ANSWER 2

7. Warm-Up Exercises EXAMPLE 2 Find a least common multiple (LCM) Find the least common multiple of 4x 2 –16 and 6x 2 –24x + 24. SOLUTION STEP 1 Factor each polynomial. Write numerical factors as products of primes. 4x 2 – 16 = 4(x 2 – 4) = (2 2 )(x + 2)(x – 2) 6x 2 – 24x + 24 = 6(x 2 – 4x + 4) = (2)(3)(x – 2) 2

8. Warm-Up Exercises EXAMPLE 2 Find a least common multiple (LCM) STEP 2 Form the LCM by writing each factor to the highest power it occurs in either polynomial. LCM = (2 2 )(3)(x + 2)(x – 2) 2 = 12(x + 2)(x – 2) 2

9. Warm-Up Exercises EXAMPLE 3 Add with unlike denominators Add: 9x29x2 7 + x 3x 2 + 3x SOLUTION To find the LCD, factor each denominator and write each factor to the highest power it occurs. Note that 9x 2 = 3 2 x 2 and 3x 2 + 3x = 3x(x + 1), so the LCD is 3 2 x 2 (x + 1) = 9x 2 (x + 1). Factor second denominator. 7 9x29x2 x 3x 2 + 3x = 7 9x29x2 + 3x(x + 1) x + LCD is 9x 2 (x + 1). 7 9x29x2 x + 1 + 3x(x + 1) x 3x3x 3x3x =

10. Warm-Up Exercises EXAMPLE 3 Add with unlike denominators Multiply. Add numerators. 3x 2 + 7x + 7 9x 2 (x + 1) = 7x + 7 9x 2 (x + 1) 3x23x2 +=

11. Warm-Up Exercises EXAMPLE 4 Subtract with unlike denominators Subtract: x + 2 2x – 2 –2x –1 x 2 – 4x + 3 – SOLUTION x + 2 2x – 2 –2x –1 x 2 – 4x + 3 – x + 2 2(x – 1) – 2x – 1 (x – 1)(x – 3) – = Factor denominators. x + 2 2(x – 1) = x – 3 – – 2x – 1 (x – 1)(x – 3) 2 2 LCD is 2(x − 1)(x − 3). x 2 – x – 6 2(x – 1)(x – 3) – 4x – 2 2(x – 1)(x – 3) – = Multiply.

12. Warm-Up Exercises EXAMPLE 4 Subtract with unlike denominators x 2 – x – 6 – (– 4x – 2) 2(x – 1)(x – 3) = Subtract numerators. x 2 + 3x – 4 2(x – 1)(x – 3) = Simplify numerator. Factor numerator. Divide out common factor. Simplify. = (x –1)(x + 4) 2(x – 1)(x – 3) x + 4 2(x –3) =

13. Warm-Up Exercises GUIDED PRACTICE for Examples 2, 3 and 4 Find the least common multiple of the polynomials. 5. 5x 3 and 10x 2 –15x LCM = 5x 3 (2x – 3) ANSWER 6. 8x – 16 and 12x 2 + 12x – 72 LCM = 24(x – 2)(x + 3) ANSWER

14. Warm-Up Exercises GUIDED PRACTICE for Examples 2, 3 and 4 4x4x 3 – 7 1 7. Perform the indicated operation and simplify. 21 – 4x 28x ANSWER 1 3x23x2 + x 9x 2 – 12x 8. ANSWER x 2 + 3x – 4 3x 2 (3x – 4)

15. Warm-Up Exercises GUIDED PRACTICE for Examples 2, 3 and 4 x x 2 – x – 12 + 5 12x – 48 9. Perform the indicated operation and simplify. ANSWER 17x + 15 12(x +3)(x − 4) x + 1 x 2 + 4x + 4 – 6 x 2 – 4 10. x 2 – 7x – 14 (x + 2) 2 (x – 2) ANSWER

16. Warm-Up Exercises EXAMPLE 6 Simplify a complex fraction (Method 2) Simplify: 5 x + 4 1 + 2 x SOLUTION The LCD of all the fractions in the numerator and denominator is x(x + 4). 5 x + 4 1 + 2 x 5 1 + 2 x = x(x+4) Multiply numerator and denominator by the LCD. x + 2(x + 4) 5x5x = Simplify. 5x5x 3x + 8 =

17. Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 x 6 x 3 – x 5 7 10 – 11. Simplify the complex fraction. – 5x 3 (2x – 7) ANSWER 2 x – 2 x + 4 3 12. 2 (1 – 2x ) 2 + 3x ANSWER

18. Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 3 x + 5 2 x – 3 + 1 x + 5 13. Simplify the complex fraction. 3(x – 3) 3x + 7 ANSWER

19. Warm-Up Exercises Daily Homework Quiz 1. Find the least common multiple of 3x 2 – 6x – 45 and 2x 2 – 20x + 50. 2. Add: 5 x 2 – 1 + 2x2x x 2 + 5x – 6 ANSWERS 6x(x – 5) 2 (x +3) ANSWERS 2x 2 +7x + 30 (x +1)(x – 1)(x + 6)

20. Warm-Up Exercises Daily Homework Quiz ANSWERS x 2 – 5x – 11 3(x – 5)(x + 2) ANSWERS 6x26x2 4x 2 – 5x + 15 Subtract: x – 4 3x – 15 – x +1 x 2 – 3x – 10 3. Simplify: 6 x – 3 4 5 x2x2 – 4.