Topic 8.2.2

1. Dividing Rational Expressions Topic 8.2.2

2. Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions California Standard: 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. What it means for you: You’ll divide rational expressions by factoring and cancelling. • Key words: • rational • reciprocal • common factor

3. Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions Dividing by rational expressions is a lot like multiplying — you just have to do an extra step first. That extra step is finding the reciprocal.

4. mv m m m b b b v ÷ = • = cb b c v v c v c That is, to divide by , multiply by the reciprocal of . Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions Dividing is the Same as Multiplying by the Reciprocal Given any nonzero expressionsm, c, b, and v:

5. 1 1 Suppose you pick a number such as 10 and divide by . 2 2 The question you’re trying to answer is… “How many times does go into 10?” …or “How many halves are in 10?” Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions Dividing is the Same as Multiplying by the Reciprocal You can extend this concept to the division of any rational expression.

6. 1 1 1 1 2 2 2 2 Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions Dividing is the Same as Multiplying by the Reciprocal Division Equivalent to 10 ÷ = 20 10 × 2 = 20 10 ÷ = 30 10 × 3 = 30 10 ÷ = 40 10 × 4 = 40 10 ÷ = 10n 10 × n = 10n So, 10 divided by a fraction is equivalent to 10 multiplied by the reciprocal of that fraction.

7. 1 a So you can alwaysrewrite an expression a ÷ b in the form a • = (where b is any nonzero expression). b b Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions Dividing is the Same as Multiplying by the Reciprocal Dividing anything by a rational expression is the same as multiplying by the reciprocal of that expression. As always, you should cancel any common factors in your answer to give a simplified fraction.

8. Example 1 Simplify ÷ (k + 5). k2 – 25 k2 – 25 k2 – 25 k2 – 25 2k 2k 2k 2k ÷ (k + 5) can be written as: ÷ 1 1 k + 5 k + 5 k + 5 1 (k – 5)(k + 5) = • = • 2k Topic 8.2.2 Dividing Rational Expressions Solution Rewrite the division as multiplication by the reciprocal of the divisor. Factor as much as you can: Solution continues… Solution follows…

9. Example 1 Simplify ÷ (k + 5). k2 – 25 k2 – 25 2k 2k 1 1 k + 5 1 k + 5 1 k – 5 k – 5 = 2k 2k (k – 5)(k + 5) (k – 5)(k + 5) = • 2k 2k • = = Topic 8.2.2 Dividing Rational Expressions Solution (continued) Cancel any common factors between the numerators and denominators. Checkyour answer. Multiply your answer by (k + 5):

10. m2 – 4 m2 – 4 m + 2 m – 1 m – 1 2m Simplify ÷ . m2 – 3m + 2 m2 – 3m + 2 m – 1 2m 2m 2m = • 1 (m + 2)(m – 2) 1 = • 1 1 (m – 2)(m – 1) = Topic 8.2.2 Dividing Rational Expressions Example 2 Solution Rewrite the division as multiplication by the reciprocal of the divisor. Factor all numerators and denominators. Cancel any common factors between the numerators and denominators. Solution follows…

11. b2c2d2 a(b – 2) a + 3 bcd2 2a abc b2 – 3b + 2 ÷ ÷ bdc3 b + 1 a – 2 abc (b + 1)(b – 1) d3 2 a2 – 9 a2 + 3a + 2 x2 – 4x – 5 a2 – 1 ÷ ÷ x – 6 a – 3 a2 + a – 6 a2 – a – 6 a2 – 4a + 3 x2 – 25 a + 3 x – 2 x2 – 5x – 6 ÷ x2 + 3x – 10 Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions Guided Practice Divide and simplify each expression. 1. 2. 3. 4. 5. 1 Solution follows…

12. d d a a a c e e f b b d b c e c f f ÷ ÷ = • ÷ = • • Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions You Can Divide Long Strings of Expressions At Once Just like multiplication, you can divide any number of rational expressions at once, but it makes a big difference which order you do things in. If there are no parentheses, you always work through the calculation from left to right, so that:

13. x2 + 5x + 6 x2 + 5x + 6 x2 + 2x + 1 x2 + 2x + 1 x2 + 5x + 6 x – 1 x2 + x – 2 2x2 + 2x 2x2 + 2x Simplify ÷ ÷ . x2 + 2x + 1 x2 + 3x x2 + 3x x2 + 3x x – 1 x – 1 x2 + x – 2 x2 + x – 2 2x2 + 2x = • ÷ = • • Topic 8.2.2 Dividing Rational Expressions Example 3 Solution Rewrite each division as a multiplication by the reciprocal of the divisor. Solution continues… Solution follows…

14. x2 + 5x + 6 x2 + 2x + 1 x2 + 5x + 6 x – 1 x2 + x – 2 2x2 + 2x Simplify ÷ ÷ . x2 + 2x + 1 x2 + 3x x2 + 3x x – 1 x2 + x – 2 2x2 + 2x 2 (x + 2)(x + 3) 2x(x + 1) x – 1 1 1 1 1 1 = • • = (x + 1)(x + 1) (x + 2)(x – 1) x(x + 3) x + 1 = • • 1 1 1 1 1 Topic 8.2.2 Dividing Rational Expressions Example 3 Solution (continued) Factor all numerators and denominators. Cancel any common factors between the numerators and denominators.

15. a a a c e b d b b f d • e c • f ÷ ÷ = ÷ = • d • e c • f Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions You Can Divide Long Strings of Expressions At Once Parentheses override this order of operations, so you need to simplify any expressions in parentheses first:

16. (x + 4) (x – 2) –k2 + 3k – 2 x2 – 4x – 12 6x3 – 36x2 (x – 2)(x + 4) –x2 + 2x + 8 k2 – 1 k2 + 5k – 6 (x + 1) (x + 3) 2x2 – 3x – 2 –2x2 + 7x + 4 2k2 – 10k 3x3 + 3x2 – 18x (x + 3) k2 – 4k – 5 ÷ ÷ – 2k2 – 14k k2 – 9k + 14 k2 + 5k – 6 x + 3 1 ÷ ÷ x + 1 2x ÷ ÷ Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions Guided Practice Divide and simplify each expression. 6. 7. 8. Solution follows…

17. a d a a c c e e e b d b d b c f f f = ÷ × ÷ × × × Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions You Can Multiply and Divide at the Same Time Say you have an expression like this to simplify: Again, you work from left to right, and anywhere you get a division, multiply by the reciprocal, so:

18. Example 4 p2 – 2pq + q2 p2 – 2pq + q2 p2 + pq – 2q2 p2 + pq – 2q2 pq + 2q2 p2 + q2 Simplify × ÷ . p2 – 2pq – 3q2 p2 – 2pq – 3q2 p2 – 3pq p2 – 3pq pq + 2q2 p2 + q2 = × × (p + 2q)(p – q) (p – q)(p + q) p(p – 3q) p 1 1 1 1 1 1 1 1 1 1 (p – q)(p – q) (p – 3q)(p + q) q(p + 2q) q = × × = Topic 8.2.2 Dividing Rational Expressions Solution Rewrite any divisions as multiplications by reciprocals. Factor all numerators and denominators. Cancel any common factors. Solution follows…

19. 2a – 2 2a2 – 7a + 3 2a2 – 7a + 3 2a2 – 7a + 3 2a2t + 12at – 14t 2a2t + at – t 2a2t + at – t Show that × ÷ ÷ = Example 5 a2 + 4a – 21 a2 + 4a – 21 a2 + 4a – 21 2a2t + 12at – 14t 2a2t + 12at – 14t a + 1 2a2t + at – t = Topic 8.2.2 Dividing Rational Expressions Justify your work. Solution The question asks you to justify your work, so make sure you can justify all your steps. Start with left-hand side Definition of division Solution continues… Solution follows…

20. 2a – 2 (2a – 1)(a – 3) (2a – 1)(a – 3) 2a2 – 7a + 3 2a2 – 7a + 3 2t(a2 + 6a – 7) 2a2t + 12at – 14t 2t(a + 7)(a – 1) 2a2t + at – t Show that × ÷ = Example 5 a2 + 4a – 21 a2 + 4a – 21 (a + 7)(a – 3) (a + 7)(a – 3) 2a2t + 12at – 14t t(2a – 1)(a + 1) a + 1 t(2a2 + a – 1) 2a2t + at – t = = = × × Topic 8.2.2 Dividing Rational Expressions Justify your work. Solution (continued) Equation carried forward Distributive property Distributive property Solution continues…

21. 2a – 2 2a – 2 (2a – 1)(a – 3) (2a – 1) (2a – 1) 2a2 – 7a + 3 t(2a – 1)(a + 7)(a – 3) 2t(a + 7)(a – 1) 2a2t + at – t Show that ÷ = Example 5 a2 + 4a – 21 (a + 7)(a – 3) 2a2t + 12at – 14t t(2a – 1)(a + 1)(a – 3) t(2a – 1)(a + 1) a + 1 a + 1 (a + 1) (a + 1) = × = × = = Topic 8.2.2 Dividing Rational Expressions Justify your work. Solution (continued) Equation carried forward Commutative and associative properties of multiplication Inverse and identity properties, and distributive property

22. t – 1 a2 – 1 x2 + 5x – 14 a2 + a – 12 a2 + 2a – 3 x2 + 2x – 3 a2 – 5a + 6 a2 + 5a + 4 x2 + 6x – 7 a2 – 2a – 3 t2 – 1 t + 1 1 a2 – 4 x2 – 4x – 21 a2 – 2a – 3 x2 – 5x + 6 –a2 + 2a + 15 a2 + a – 2 a2 – 3a – 10 a2 + 2a + 1 x2 – 6x – 7 ÷ × t2 + 2t – 3 t2 + 4t + 3 ÷ × x + 1 a + 3 a – 1 a + 3 a + 2 x – 3 ÷ × ÷ × – Lesson 1.1.1 Topic 8.2.2 Dividing Rational Expressions Guided Practice Simplify these rational expressions. 9. 10. 11. 12. t2 – 1 Solution follows…

23. b3 – 4b –x3 – 3x2 – 2x x2 – 6x + 8 y2 – y – 2 t2 + 2t – 3 k2 – m2 b2 – b – 2 x2 – x – 6 3t – 3 –y + 2 2k2 + 2m –x3 + x y2 + 3y – 4 2k2 + km – m2 b3 + b t2 + 4t + 3 x2 – 2x – 3 x2 – 4 x3 – 2x2 – 3x t2 – t – 2 2k2 + 3km – 2m2 y2 – 3y + 2 –2x2 + 4x + 16 b4 – 1 k2 + km – 2m2 2x2 – 16x + 32 y2 – y – 2 ÷ ÷ – 2k + 2m y + 4 x3 – x t – 2 ÷ ÷ 3 x3 + x2 – x – 3 ÷ ÷ Topic 8.2.2 Dividing Rational Expressions Independent Practice Divide and simplify each expression. 1. 2. 3. 4. 5. 6. b2 + b – 2 Solution follows…

24. x2 – 3x + 2 (m – v)2 a3 – 4a b2 – 1 b2 – 2b + 1 a2 + a – 2 m2 – 3mv + 2v2 –x + 2 b2 – 2b – 3 –a2 + 2a m2 – v2 x2 + x – 2 b2 – 4b + 3 a2 – a – 2 x2 – 3x – 10 (m – 2v)2 m – 2v a2 – a – 2 ÷ ÷ – m + v a – 1 ÷ ÷ Topic 8.2.2 Dividing Rational Expressions Independent Practice Divide and simplify each expression. 7. 8. 9. 10. 1 –x + 5 Solution follows…

25. 1 t + 2 2x2 – 5x – 12 t2 – t – 6 x2 – 9 y + 5 y2 + 4y – 5 x2 – 16 t + 3 y2 – 4y – 5 t2 + 6t + 9 y2 – 6y + 5 x2 + 2x – 8 4x2 + 8x + 3 2x2 + 7x + 3 y + 1 x – 2 1 ÷ ÷ (t + 2)2 x – 3 ÷ ÷ ÷ (t2 – 4) ÷ Topic 8.2.2 Dividing Rational Expressions Independent Practice Divide and simplify each expression. 11. 12. 13. 1 Solution follows…

26. k2 – 5k + 6 –2k2 – 6k – 4 k2 + 3k + 2 2m – 5n (m + n)2(2m + n)2(–2m + 5n) • ÷ 4m + n (4m + n)(m – 2n)2(–m + n)2 k2 + 2k – 8 k2 – 2k – 3 k2 + 5k + 4 –2v2 + 4vw v2 – w2 v3 – vw2 –2v2 + 5vw – 3w2 × ÷ × 3v2 – 4vw + w2 –2vw + 3w2 6v2 + 4vw2 – 2w3 –4v2 + 4vw + 8w2 –m2 + 3mn – 2n2 –m2 + 3mn – 2n2 m2 + 2mn + n2 m2 + 2mn + n2 4m2 + 5mn + n2 2m2 + 3mn + n2 2m2 + 3mn + n2 4m2 + 5mn + n2 ÷ ÷ ÷ –m2 + 3mn – 2n2 –m2 + 3mn – 2n2 2m2 – 5mn – 3n2 2m2 – 5mn – 3n2 –2m2n + 5mn2 –2m2n + 5mn2 m2n– 3mn2 m2n– 3mn2 – ÷ ÷ ÷ Topic 8.2.2 Dividing Rational Expressions Independent Practice Simplify these rational expressions. 14. 15. 16. 17. –2 1 Solution follows…

27. Topic 8.2.2 Dividing Rational Expressions Round Up It’s really important that you can justify your work step by step, because division of rational expressions can involve lots of calculations that look quite similar.