1. Concept 5
Rational expressions

2. Rational Expressions
Rational Expression - the ratio of _____ polynomials. (Like a _____________ with a polynomial in the ________________ and the ______________________.
 Examples: Non-Examples:
 Characteristics: 1. 2. 2
fraction
numerator
denominator

3. Factor top & bottom:
Cancel & Reduce if possible:

4. Factor top & bottom:
Cancel & Reduce if possible:

5. Examples:
𝑥 2 15𝑥
𝑥 2 4𝑥+4
3. 𝑥 2 −4 𝑥+2
𝑥 2 +21𝑥 42𝑥 4

6. 𝑥 3 + 7𝑥 2 6 𝑥 5 + 9𝑥 3
6. 𝑥 2 +9𝑥+18 4𝑥+24
7. 𝑥 2 +6𝑥−16 𝑥 2 +10𝑥−24
8. 𝑥 3 −𝑥 𝑥 3 − 7𝑥 2 +6𝑥
𝑥 𝑥 𝑥 2 −30 𝑥 4

7. Solving Polynomials
Concept 6

8. To solve a polynomial, factor completely
To solve a polynomial, factor completely. Any terms can then be set equal to zero and solve for x.
2. 6𝑥 2 +6𝑥=0
1. 𝑥 2 −𝑥 −42=0
6𝑥 𝑥+1 =0
𝑥−7 𝑥+6 =0
6𝑥=0
𝑥+1=0
𝑥−7=0
𝑥+6=0
𝑥=0
𝑥=−1
𝑥=7
𝑥=−6
3. 5𝑥 2 +37𝑥= −14
5𝑥 2 +37𝑥+14=0
5𝑥+2=0
5𝑥=−2
𝑥 2 +37𝑥+70=0
𝑥= −2 5
𝑥+2 𝑥+35 =0 5 5
5𝑥+2 𝑥+7 =0
𝑥+7=0
𝑥=−7

9. 4. 7 𝑥 2 +53𝑥+28=0 5. 2𝑥 2 −4𝑥−3=−5 𝑥 2 +53𝑥+196=0 2𝑥 2 −4𝑥+2=0
4. 7 𝑥 2 +53𝑥+28=0
𝑥 2 +53𝑥+196=0
5. 2𝑥 2 −4𝑥−3=−5
𝑥+49 𝑥+4 =0
2𝑥 2 −4𝑥+2=0 7 7
2 𝑥 2 − 2𝑥+1 =0
𝑥+7 7𝑥+4 =0
2 𝑥−1 𝑥−1 =0
𝑥−1 =0
𝑥+7=0
7𝑥+4=0
𝑥=1
7𝑥=−4
𝑥=−7
𝑥= −4 7

10. Quadratic Formula – used to solve equations when it can’t be factored.
Before substituting in values for a, b, and c, rearrange the equation to look like. 𝑎 𝑥 2 +𝑏𝑥+𝑐=0
The quadratic formula is used to solve for values that make the equation true and are not factorable.

11. 𝑥= − ± −
𝑥= − −8 ± −8 2 −4 1 −
So a = = 1
b = = -8
𝑥= 8 ±
c = = -48
𝑥= 8 ± 1
-48 -8 -8
𝑥= 8 ±16 2 𝑥=
𝑥= 8 −16 2
𝑥= −8 2 =4 1
𝑥= 24 2 =12

12. So a = = 6
b = = 9
c = = -18
𝑥= − ± −
x= − 9 ± −4 6 −
𝑥= −9 ±
𝑥= −9 ±

13. Concept 7: evaluating and finding zeros of rational expressions.

14. Evaluating Rational Expressions
Evaluating—Substituting a specific value in for x into a rational expressions and evaluating to find a specific answer.
Evaluate 5 −𝑥 2𝑥+4 when x = 1
5 −(1) 2(1)+4
= 4 2+4
= 4 6
= 2 3

15. 5 −(3) 2(3)+4
= 2 6+4
= 2 10
= 1 5
5 −(−2) 2(−2)+4
= 7 −4+4
= 7 0
= undefined

16. Evaluate 𝑥 2 −16 2𝑥+7 at the given values of x.
a. x = 1 b. x = -5 c. x = 4
𝑥 2 −16 2𝑥+7
= (1) 2 −16 2(1)+7
= (−5) 2 −16 2(−5)+7
= (4) 2 −16 2(4)+7
= 1−16 2+7
= 25−16 −10+7
= 16−16 8+7
= −15 9
= 9 −3
= 0 15
= −5 3 or −1 2 3
=−3 =0

17. Finding zeros— finding the values of the variable that makes the denominator of a rational expression zero.
Find the values that make the denominator zero.

18. Finding zeros— finding the values of the variable that makes the denominator of a rational expression zero.

20. Concept 8: Multiplying Rational Expressions

21. Multiplying Rational Numbers
Step 1:
Factor both the _________________ and the _______________
Step 2:
Write together as one fraction, by ____________ _______.
Step 3 :
___________ the expression.
Step 4:
Multiply _________ remaining factors in the _______________ or the ____________________.
numerator
denominator
multiplying
across
Simplify
any
numerator
denominator

22. 1 1.
= 12 𝑥 2 𝑥 3 24𝑦
= 1 𝑥 5 2𝑦 2
= 2 𝑎 2 𝑏 2 𝑎 𝑏 2 𝑐
= 2𝑎 𝑐 2.

23. = 4 3𝑥 = 2 𝑥−6 ∗14 7𝑥∗3(𝑥−6) 3. = 𝑥 3𝑥+1 6(𝑥−5) = 3𝑥 2 +𝑥 6𝑥−30 4. 2
2(x – 6) 2
= 2 𝑥−6 ∗14 7𝑥∗3(𝑥−6)
= 4 3𝑥 3. 1
3(x – 6)
= 𝑥 3𝑥+1 6(𝑥−5)
= 3𝑥 2 +𝑥 6𝑥−30 4.

24. = 16 𝑥 2 𝑥−2 (𝑥+5) 8 𝑥 6 (𝑥+5) 5. = 2 𝑥−2 𝑥 4 = 2𝑥−4 𝑥 4
(x - 2)(x + 5) 2
= 16 𝑥 2 𝑥−2 (𝑥+5) 8 𝑥 6 (𝑥+5) 5. 1
𝑥 4
= 2 𝑥−2 𝑥 4
= 2𝑥−4 𝑥 4
5(x + 9) x
= 15 𝑥 3 𝑥+9 𝑥 2 (𝑥+9)(𝑥 −9)
= 15𝑥 𝑥−9 6.
(x + 9)(x - 9)

25. = 𝑥+2 𝑥 2 +𝑥−6 = 𝑥+2 (𝑥−2)(𝑥+3) 7. 2𝑥 2 −6𝑥−56 𝑥−3 ∙ 𝑥 2 −9 𝑥 2 +5𝑥+6
(x + 2)(x - 6)
= 𝑥+2 𝑥 2 +𝑥−6 7.
= 𝑥+2 (𝑥−2)(𝑥+3)
(x - 2)(x + 2)
(x - 3)(x - 6)
2(𝑥−7)(𝑥+4)
2( 𝑥 2 −3𝑥−28)
(𝑥−3)(𝑥+3)
2𝑥 2 −6𝑥−56 𝑥−3 ∙ 𝑥 2 −9 𝑥 2 +5𝑥+6
= 2(𝑥−7)(𝑥+4) (𝑥+2) 8.
(𝑥+2)(𝑥+3)
= 2( 𝑥 2 −10𝑥+4𝑥−28) 𝑥+2
= 2 𝑥 2 −12𝑥−56 𝑥+2

26. (x - 7)(x + 7)
(x - 7)(x - 3)
= (𝑥−7)(𝑥−7) 𝑥(𝑥+4)(𝑥+3)
9. 𝑥 2 −49 𝑥 𝑥 2 +28𝑥 ∙ 𝑥 2 −10𝑥+21 𝑥 2 −9
(x - 3)(x + 3)
𝑥( 𝑥 2 +11𝑥+28)
= 𝑥 2 −14𝑥+49 𝑥( 𝑥 2 +7𝑥+12)
x(x + 4)(x + 7)
= 𝑥 2 −14𝑥+49 𝑥 3 +7 𝑥 2 +12𝑥

27. = 𝑥(𝑥+11) 4(𝑥−2) 10. = 𝑥 2 +11𝑥 4𝑥−8 = 5𝑥(𝑥+7) 𝑥+5 11. = 5𝑥 2 +35𝑥 𝑥+5
(x + 11)(x + 11)
= 𝑥(𝑥+11) 4(𝑥−2)
10. 4
2x(x - 2)
= 𝑥 2 +11𝑥 4𝑥−8 5x
(x + 7)(x - 7)
= 5𝑥(𝑥+7) 𝑥+5
11. 1
= 5𝑥 2 +35𝑥 𝑥+5
2( 𝑥 2 −2𝑥−35)
2(𝑥−7)(𝑥+5)

28. Concept 9: Dividing Rational Expressions
We ___________ divide rational expressions we ______________ by the ________________ of the ___________ expression.
do not
multiply
reciprocal
last

29. 1. = 𝑟 3 𝑠 𝑡 ∙ 𝑡 3 𝑟 𝑠 3 = 𝑟 2 𝑡 2 𝑠 2 = 𝑢 5 𝑥 𝑦 ∙ 𝑦 𝑢 𝑥 2 = 𝑢 4 𝑥 2.
= 𝑟 3 𝑠 𝑡 ∙ 𝑡 3 𝑟 𝑠 3
= 𝑟 2 𝑡 2 𝑠 2
𝑠 2 1 1
𝑢 4 1 1
= 𝑢 5 𝑥 𝑦 ∙ 𝑦 𝑢 𝑥 2
= 𝑢 4 𝑥 2. x 1 1

30. 3. = 𝑚 5 𝑛 𝑝 ∙ 𝑝 4 𝑚 𝑛 4 = 𝑚 4 𝑝 3 𝑛 3 = 30 𝑦 2 +4𝑦−12 ∙ 𝑦−2 6𝑦
= 𝑚 5 𝑛 𝑝 ∙ 𝑝 4 𝑚 𝑛 4
= 𝑚 4 𝑝 3 𝑛 3 1
𝑛 3 1 5 1
= 30 𝑦 2 +4𝑦−12 ∙ 𝑦−2 6𝑦
= 5 𝑦+6 4. 1
𝑦−2 (𝑦+6) 1

31. 5. 3 1
= 15 𝑦 2 +2𝑦−8 ∙ 𝑦−2 5𝑦
= 3 𝑦(𝑦+4) 𝑦
(𝑦−2)(𝑦+4)
= 3 𝑦 2 +4𝑦 6.
(𝑦+7)(𝑦−4)
(𝑦+2)(𝑦−7)
= 𝑥 2 +3𝑥−28 𝑥 2 +4𝑥+4 ∙ 𝑥 2 −5𝑥−14 𝑥 2 −49
(𝑦+2)(𝑦+2)
(𝑦+7)(𝑦−7)
= 𝑦−4 𝑦+2

32. 7. = 𝑦 2 −9 𝑦 2 ∙ 𝑦+2 𝑦 5 +3 𝑦 4 = (𝑦−3)(𝑦−2) 𝑦 6 8.
(𝑦+3)(𝑦−3)
= 𝑦 2 −9 𝑦 ∙ 𝑦+2 𝑦 5 +3 𝑦 4
= (𝑦−3)(𝑦−2) 𝑦 6
𝑦 4 (𝑦+3)
4( 𝑥 2 −16) 8.
x(𝑥−4)
= 4𝑥 2 −64 8𝑥 ∙ 𝑥 2 −4𝑥 𝑥 2 −16
= 4(𝑥−4) 8
= 𝑥−4 2 2