1. 6.1 – Rational Expressions and Functions: Multiplying and Dividing
Math 71A
6.1 – Rational Expressions and Functions: Multiplying and Dividing

2. A rational expression is a _________________ divided by a _________________. ex: 50𝑥 10−𝑥 3𝑥+1 𝑥 2 −3𝑥+2 𝑥 2 +3𝑥𝑦−10 𝑦 2 3 𝑥 2 −7𝑥𝑦+2 𝑦 2

3. A rational expression is a _________________ divided by a _________________. ex: 50𝑥 10−𝑥 3𝑥+1 𝑥 2 −3𝑥+2 𝑥 2 +3𝑥𝑦−10 𝑦 2 3 𝑥 2 −7𝑥𝑦+2 𝑦 2
polynomial

4. A rational expression is a _________________ divided by a _________________. ex: 50𝑥 10−𝑥 3𝑥+1 𝑥 2 −3𝑥+2 𝑥 2 +3𝑥𝑦−10 𝑦 2 3 𝑥 2 −7𝑥𝑦+2 𝑦 2
polynomial
polynomial

5. A rational expression is a _________________ divided by a _________________. ex: 50𝑥 10−𝑥 3𝑥+1 𝑥 2 −3𝑥+2 𝑥 2 +3𝑥𝑦−10 𝑦 2 3 𝑥 2 −7𝑥𝑦+2 𝑦 2
polynomial
polynomial

6. Ex 1. Find the domain of 𝑓 𝑥 = 𝑥−5 2 𝑥 2 +5𝑥−3 Ex 2
Ex 1. Find the domain of 𝑓 𝑥 = 𝑥−5 2 𝑥 2 +5𝑥−3 Ex 2. Find the domain of 𝑓 𝑥 = 3 𝑥 2 +1

7. To simplify a rational expression: 1
To simplify a rational expression: 1. ___________ top and bottom completely. 2. __________ any common factors.

8. To simplify a rational expression: 1
To simplify a rational expression: 1. ___________ top and bottom completely. 2. __________ any common factors.
Factor

9. To simplify a rational expression: 1
To simplify a rational expression: 1. ___________ top and bottom completely. 2. __________ any common factors.
Factor
Cancel

10. Ex 3. Simplify: 𝑥 2 +7𝑥+10 𝑥+2

11. Note: When we simplify rational expressions, we’re changing the _______ of the related rational functions. ex: The domain of 𝑥 2 +7𝑥+10 𝑥+2 is all real #’s except −2, but the domain of 𝑥+5 is all real #’s. So, to simplify, but also keep the original domain, we can write: 𝑥 2 +7𝑥+10 𝑥+2 =𝑥+5, for 𝑥≠−2

12. Note: When we simplify rational expressions, we’re changing the _______ of the related rational functions. ex: The domain of 𝑥 2 +7𝑥+10 𝑥+2 is all real #’s except −2, but the domain of 𝑥+5 is all real #’s. So, to simplify, but also keep the original domain, we can write: 𝑥 2 +7𝑥+10 𝑥+2 =𝑥+5, for 𝑥≠−2
domain

13. Note: When we simplify rational expressions, we’re changing the _______ of the related rational functions. ex: The domain of 𝑥 2 +7𝑥+10 𝑥+2 is all real #’s except −2, but the domain of 𝑥+5 is all real #’s. So, to simplify, but also keep the original domain, we can write: 𝑥 2 +7𝑥+10 𝑥+2 =𝑥+5, for 𝑥≠−2
domain

14. Note: When we simplify rational expressions, we’re changing the _______ of the related rational functions. ex: The domain of 𝑥 2 +7𝑥+10 𝑥+2 is all real #’s except −2, but the domain of 𝑥+5 is all real #’s. So, to simplify, but also keep the original domain, we can write: 𝑥 2 +7𝑥+10 𝑥+2 =𝑥+5, for 𝑥≠−2
domain

15. Note: When we simplify rational expressions, we’re changing the _______ of the related rational functions. ex: The domain of 𝑥 2 +7𝑥+10 𝑥+2 is all real #’s except −2, but the domain of 𝑥+5 is all real #’s. So, to simplify, but also keep the original domain, we can write: 𝑥 2 +7𝑥+10 𝑥+2 =𝑥+5, for 𝑥≠−2
domain

16. Ex 4. Simplify: 3 𝑥 2 +9𝑥𝑦−12 𝑦 2 9 𝑥 3 −9𝑥 𝑦 2

17. To multiply rational expressions: 1
To multiply rational expressions: 1. ___________ tops and bottoms completely. 2. __________ common factors. 3. __________ remaining factors on top and on bottom (you can leave top/bottom factored).

18. To multiply rational expressions: 1
To multiply rational expressions: 1. ___________ tops and bottoms completely. 2. __________ common factors. 3. __________ remaining factors on top and on bottom (you can leave top/bottom factored).
Factor

19. To multiply rational expressions: 1
To multiply rational expressions: 1. ___________ tops and bottoms completely. 2. __________ common factors. 3. __________ remaining factors on top and on bottom (you can leave top/bottom factored).
Factor
Cancel

20. To multiply rational expressions: 1
To multiply rational expressions: 1. ___________ tops and bottoms completely. 2. __________ common factors. 3. __________ remaining factors on top and on bottom (you can leave top/bottom factored).
Factor
Cancel
Multiply

21. To multiply rational expressions: 1
To multiply rational expressions: 1. ___________ tops and bottoms completely. 2. __________ common factors. 3. __________ remaining factors on top and on bottom (you can leave top/bottom factored).
Factor
Cancel
Multiply
(4. And of course, always simplify!)

22. Ex 5. Multiply: 𝑥+4 𝑥−7 ∙ 𝑥 2 −4𝑥−21 𝑥 2 −16 Ex 6
Ex 5. Multiply: 𝑥+4 𝑥−7 ∙ 𝑥 2 −4𝑥−21 𝑥 2 −16 Ex 6. Multiply: 4𝑥+8 6𝑥−3 𝑥 2 ∙ 3 𝑥 2 −4𝑥−4 9 𝑥 2 −4

23. Dividing is the same as multiplying by the ________________.

24. Dividing is the same as multiplying by the ________________.
reciprocal

25. Ex 7. Divide: 𝑥 2 −𝑥−12 5𝑥 ÷ 𝑥 2 −10𝑥+24 𝑥 2 −6𝑥 Ex 8
Ex 7. Divide: 𝑥 2 −𝑥−12 5𝑥 ÷ 𝑥 2 −10𝑥+24 𝑥 2 −6𝑥 Ex 8. Divide: (9 𝑥 2 −49)÷ 3𝑥−7 9