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Section 5.5

Section 5.5. Adding and Subtracting Rational Expressions. HW Quiz. Add the following fractions. Complex Fractions. 2 x. 5. b. –. x + 6. x + 6. 7 + 3. 10. 5. 7. 7. 3. 3. =. =. =. a. a. 4 x. +. +. 4 x. 2 x. 4 x. 4 x. 4 x. 4 x. 2 x. 5. 2 x – 5. b. –. =.

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Section 5.5

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  1. Section 5.5 Adding and Subtracting Rational Expressions

  2. HW Quiz

  3. Add the following fractions

  4. Complex Fractions

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

  6. 3. 2. 1. + − – 2 4x 7 5 x 1 1 1 6x x2 x–2 12x 3x2 3x2 x–2 12x 3x x – 2 2 2x2 4. + x2+1 x2+1 for Example 1 GUIDED PRACTICE Perform the indicated operation and simplify. ANSWER ANSWER ANSWER ANSWER 2

  7. EXAMPLE 2 Find a least common multiple (LCM) Find the least common multiple of 4x2 –16 and 6x2 –24x + 24. SOLUTION STEP 1 Factor each polynomial. Write numerical factors asproducts of primes. 4x2 – 16 = 4(x2 – 4) = (22)(x + 2)(x – 2) 6x2 – 24x + 24 = 6(x2 – 4x + 4) = (2)(3)(x – 2)2

  8. 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 = (22)(3)(x + 2)(x – 2)2 = 12(x + 2)(x – 2)2

  9. x + 1 7 x x + 1 + 9x2 3x2 + 3x 7 7 x x = + + 9x2 9x2 3x2 + 3x 3x(x + 1) 7 3x x 9x2 3x + 3x(x + 1) EXAMPLE 3 Add with unlike denominators Add: SOLUTION To find the LCD, factor each denominator and write each factor to the highest power it occurs. Note that9x2 = 32x2and3x2 + 3x = 3x(x + 1), so the LCD is 32x2 (x + 1) = 9x2(x + 1). Factor second denominator. = LCD is 9x2(x + 1).

  10. 3x2 7x + 7 = + 9x2(x + 1) 9x2(x + 1) 3x2 + 7x + 7 = 9x2(x + 1) EXAMPLE 3 Add with unlike denominators Multiply. Add numerators.

  11. x – 3 x + 2 x + 2 –2x –1 –2x –1 – x– 3 2x – 2 2x – 2 x2 – 4x + 3 x2 – 4x + 3 – x + 2 x + 2 – 2x – 1 – 2x – 1 – = (x – 1)(x – 3) (x – 1)(x – 3) 2(x – 1) 2(x – 1) – = 2 x2 – x – 6 – 4x – 2 – 2 = 2(x – 1)(x – 3) 2(x – 1)(x – 3) EXAMPLE 4 Subtract with unlike denominators Subtract: SOLUTION Factor denominators. LCD is 2(x − 1)(x − 3). Multiply.

  12. x2 – x – 6 – (– 4x – 2) = 2(x – 1)(x – 3) x2 + 3x – 4 = 2(x – 1)(x – 3) (x –1)(x + 4) = 2(x – 1)(x – 3) x + 4 = 2(x –3) EXAMPLE 4 Subtract with unlike denominators Subtract numerators. Simplify numerator. Factor numerator. Divide out common factor. Simplify.

  13. for Examples 2, 3 and 4 GUIDED PRACTICE Find the least common multiple of the polynomials. 5. 5x3and 10x2–15x ANSWER LCM = 5x3 (2x – 3) 6. 8x – 16 and 12x2 + 12x – 72 ANSWER LCM = 24(x – 2)(x + 3)

  14. x 1 8. + x2 + 3x – 4 3x2 9x2 – 12x 3 1 7. 3x2 (3x – 4) – 4x 7 21 – 4x 28x for Examples 2, 3 and 4 GUIDED PRACTICE Perform the indicated operation and simplify. ANSWER ANSWER

  15. x 5 9. + x2 – x – 12 12x – 48 x2 – 7x –14 17x + 15 (x + 2)2 (x – 2) 12(x +3)(x − 4) x + 1 6 10. – x2 + 4x + 4 x2 – 4 for Examples 2, 3 and 4 GUIDED PRACTICE Perform the indicated operation and simplify. ANSWER ANSWER

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

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

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

  19. Find the least common multiple of 3x2 – 6x – 45 and 2x2 – 20x + 50. 1. 6x(x – 5)2 (x +3) ANSWERS 5 2x 2. Add: + x2 + 5x – 6 x2 – 1 2x2 +7x + 30 ANSWERS (x +1)(x– 1)(x + 6) Daily Homework Quiz

  20. x +1 x – 4 Subtract: – 3. 3x – 15 x2 – 3x – 10 x2 – 5x – 11 ANSWERS 3(x – 5)(x + 2) 6 x – 3 Simplify: 4. 5 4 – x – 3 x2 6x2 4x2 – 5x + 15 ANSWERS Daily Homework Quiz

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