Algebra I Honors: Quarter 4 Exam Review

ANSWER KEY 10-1 Pythagorean Theorem Determine if each set of lengths can be the side lengths of a right triangle.

1. 12, 60, 61

122 + 602 = 612

144 + 3600 = 37213744 ≠ 3721

No

2. 15, 36, 39

152 + 362 = 392

225 +1296 = 15211521= 1521

Yes

3. 12, 35, 38

122 + 352 = 382

144 +1225 = 14441369 ≠ 1444

No

10-2 Simplifying Radicals Simplify each radical expression. (Rationalize the denominator, if applicable.)

4. 192s2

64s2 ⋅3 = 8s 3

5. 3 150b8

3 25b8 ⋅6 = 3⋅5b4 6 =

15b4 6

6. 20x2y3

4x2y2 ⋅5y = 2xy 5y

7. 3x3

64x2

3x64

=3x8

8. 2 2448t 4

22 ⋅ t 4

= 2t 2 2

= 2 22t 2

=2t 2

9. 3xy17

507x5y9

y8

169x4=

y4

13x2

10. 4 10 i 2 90 4 ⋅2 ⋅ 10 ⋅90 = 8 900 = 8 ⋅30 =

240

11. 3 5c i 7 15c2

3⋅7 ⋅ 5 ⋅15 ⋅c3 =

21 25c2 ⋅3c = 21⋅5c 23c = 105c 3c

12. −6 15n5 i 2 75

−6 ⋅2 ⋅ 15 ⋅75 ⋅n5 =

−12 225n4 ⋅5n = −12 ⋅15n2 5n = −180n2 5n

? ?

? ?

?

?

13. 27

27⋅ 77=2 77

14. 58x 5

4 ⋅2x= 52 2x

⋅ 2x2x

=

10x2 ⋅2x

=10x4x

15. 3 615

3 615

⋅ 1515

= 3 9 ⋅1015

=

3⋅3 1015

=3 105

10-3 Operations with Radical Expressions Simplify each expression. (Rationalize the denominator, if applicable.)

16. 52 −1 52 −1

⋅ 2 +12 +1

=

5( 2 +1)( 2)2 − (1)2

=

5 2 + 52 −1

=

5 2 + 5

17. 37 − 3 3

7 − 3⋅ 7 + 37 + 3

=

3( 7 + 3)( 7)2 − ( 3)2

=

3 7 + 3 37 − 3

=

3 7 + 3 34

18. −26 + 11 −26 + 11

⋅ 6 − 116 − 11

=

−2( 6 − 11)( 6)2 − ( 11)2

=

−(2 6 − 2 11)6 −11

=

−(2 6 − 2 11)−5

=

2 6 − 2 115

19. 12 + 4 75 − 36 4 ⋅3 + 4 25 ⋅3 − 36 = 2 3 + 4 ⋅5 3 − 6 = 2 3 + 20 3 − 6 =

22 3 − 6

20. 18 + 3 72 + 4 9 ⋅2 + 3 36 ⋅2 + 4 = 3 2 + 3⋅6 2 + 2 = 3 2 +18 2 + 2 =

21 2 + 2

21. 2 700 − 3 20 + 5 28 2 100 ⋅7 − 3 4 ⋅5 + 5 4 ⋅7 =

2 ⋅10 7 − 3⋅2 5 + 5 ⋅2 7 = 20 7 − 6 5 +10 7 =

30 7 − 6 5

22. (3 11 + 7)2 (3 11)2 + 2(3 11)( 7)+ ( 7 )2 =

(9 ⋅11)+ (2 ⋅3 11⋅7)+ (7) =

99 + 6 77 + 7 =

106 + 6 77

23. (2 + 10)(4 − 12) (2 ⋅4)− (2 12)+ (4 10)− ( 10 ⋅12) =

8 − 2 4 ⋅3 + 4 10 − 4 ⋅30 = 8 − 2 ⋅2 3 + 4 10 − 2 30 = 8 − 4 3 + 4 10 − 2 30

24. (3+ 6)(5 − 12) (3⋅5)+ (5 6)− (3 12)− ( 6 ⋅12) =

15 + 5 6 − 3 4 ⋅3 − 36 ⋅2 = 15 + 5 6 − 3⋅2 3 − 6 2 = 15 + 5 6 − 6 3 − 6 2

11-1 Simplifying Rational Expressions Simplify each expression. State any excluded values.

25. n2 + 7n +12n2 + 6n + 8 (n + 3)(n + 4)(n + 2)(n + 4)

=

n + 3n + 2

; n ≠ −4,−2

26. c2 − 6c + 8c2 + c − 6 (c − 4)(c − 2)(c + 3)(c − 2)

=

c − 4c + 3

; n ≠ −3,2

27. w2 + 7w

w2 − 49 w(w + 7)

(w − 7)(w + 7)=

w

w − 7; w ≠ −7,7

11-2 Multiplying and Dividing Rational Expressions Multiply. State any excluded values. (Distribute. Do NOT leave as multiplication problems.)

28.

6y2

5i2

y + 3 6 ⋅2 ⋅ y2

5(y + 3)=

12y2

5y +15; y ≠ −3

29.

2xx +1

ix −13

2x(x −1)3(x +1)

=

2x2 − 2x3x + 3

; x ≠ −1

30.

m − 4m + 4

im

m −1 m(m − 4)

(m + 4)(m −1)=

m2 − 4mm2 + 4m −1m − 4

=

m2 − 4mm2 + 3m − 4

; m ≠ −4,1

Multiply. (Do NOT distribute. Factor and simplify, but leave the parenthesis.)

31.

4c2c + 2

ic2 + 3c + 2c −1

4c2(c +1)

⋅ (c +1)(c + 2)(c −1)

=

2 ⋅2c2(c +1)

⋅ (c +1)(c + 2)(c −1)

=

2c(c + 2)c −1

32.

b2 + 4b + 42b2 − 8

i3b − 64b

(b + 2)(b + 2)2(b + 2)(b − 2)

⋅ 3(b − 2)4b

=

(b + 2)(b + 2)2(b + 2)(b − 2)

⋅ 3(b − 2)4b

=

3(b + 2)8b

33.

t 2 − t −12t +1

it +1t + 3

(t − 4)(t + 3)(t +1)

⋅ (t +1)(t + 3)

=

(t − 4)(t + 3)(t +1)

⋅ (t +1)(t + 3)

=

t − 4

Divide. (Do NOT distribute. Factor and simplify, but leave the parenthesis.)

34. x2 + 6x + 8x2 + x − 2

÷ x + 42x + 4

(x + 2)(x + 4)(x + 2)(x −1)

÷ (x + 4)2(x + 2)

=

(x + 2)(x + 4)(x + 2)(x −1)

⋅ 2(x + 2)(x + 4)

=

2(x + 2)x −1

35. 2n2 − 5n − 34n2 −12n − 7

÷ 4n + 52n − 7

(2n +1)(n − 3)(2n +1)(2n − 7)

÷ (4n + 5)(2n − 7)

=

(2n +1)(n − 3)(2n +1)(2n − 7)

⋅ (2n − 7)(4n + 5)

=

n − 34n + 5

36.

4b −1b2 + 2b +112b − 3b2 −1 4b −1

b2 + 2b +1÷ 12b − 3b2 −1

=

(4b −1)(b +1)(b +1)

÷ 3(4b −1)(b +1)(b −1)

=

(4b −1)(b +1)(b +1)

⋅ (b +1)(b −1)3(4b −1)

=

b −13(b +1)

37.

g + 23g −1g2 + 2g6g + 2 g + 23g −1

÷ g2 + 2g6g + 2

=

(g + 2)(3g −1)

÷ g(g + 2)2(3g +1)

=

(g + 2)(3g −1)

⋅ 2(3g +1)g(g + 2)

=

2(3g +1)g(3g −1)

38.

c + 4c2 + 5c + 63c2 +12c2c2 + 5c − 3 c + 4

c2 + 5c + 6÷ 3c2 +12c2c2 + 5c − 3

=

(c + 4)(c + 2)(c + 3)

÷ 3c(c + 4)(2c −1)(c + 3)

=

(c + 4)(c + 2)(c + 3)

⋅ (2c −1)(c + 3)3c(c + 4)

=

2c −13c(c + 2)

11-3 Dividing Polynomials Divide. (Write your remainder as a fraction, using the divisor as the denominator.)

39. (−4q2 − 22q +12)÷ (2q +1)

2q +1 −4q2 − 22q +12

−2q −10

– (-4q2 – 2q) -20q + 12 – (-20q – 10) 22

−2q −10 + 222q +1

40. (2w3 + 3w −15)÷ (w −1)

w −1 2w3 + 0w2 + 3w −15

2w2 + 2w + 5

– (2w3 – 2w2) 2w2 + 3w – (2w2 – 2w) 5w – 15 – (5w – 5) -10

2w2 + 2w + 5 − 10w −1

11-4 Adding and Subtracting Rational Expressions Add or subtract. (Do NOT distribute the denominators. Simplify and leave the parenthesis.)

41. 3

b − 3− bb − 3 3− bb − 3

= -1

42. 5c2c + 7

+ c − 282c + 7

5c + c − 282c + 7

=6c − 282c + 7

43. 12 − b

− 42 − b

1− 42 − b

= − 32 − b

OR 3

b − 2

44. a

a + 3− 4a + 5

a(a + 3)

⋅ (a + 5)(a + 5)

− 4(a + 5)

⋅ (a + 3)(a + 3)

=

a(a + 5)− 4(a + 3)(a + 3)(a + 5)

=

(a2 + 5a)− (4a +12)(a + 3)(a + 5)

=

a2 + a −12(a + 3)(a + 5)

OR (a + 4)(a − 3)(a + 3)(a + 5)

45. 9

m + 2+ 8m − 7

9(m + 2)

⋅ (m − 7)(m − 7)

+ 8(m − 7)

⋅ (m + 2)(m + 2)

=

9(m − 7)+ 8(m + 2)(m + 2)(m − 7)

=

(9m − 63)+ (8m +16)(m + 2)(m − 7)

=

17m − 47(m + 2)(m − 7)

46. p

p + 3+ p + 5

4 p

(p + 3)⋅ 44+ (p + 5)

4⋅ (p + 3)(p + 3)

=

4 p + (p + 5)(p + 3)4(p + 3)

=

4 p + (p2 + 3p + 5p +15)4(p + 3)

=

p2 +12p +154(p + 3)

11-5 Solving Rational Equations

47. d

d + 3= 2dd − 3

−1

d(d + 3)

⋅ (d − 3)(d − 3)

= 2d(d − 3)

⋅ (d + 3)(d + 3)

− 11⋅ (d + 3)(d − 3)(d + 3)(d − 3)

d(d − 3)(d + 3)(d − 3)

= 2d(d + 3)(d + 3)(d − 3)

− 1(d + 3)(d − 3)(d + 3)(d − 3)

d(d − 3) = 2d(d + 3)−1(d + 3)(d − 3)

(d 2 − 3d) = (2d 2 + 6d)− (d 2 + 3d − 3d − 9)

d 2 − 3d = 2d 2 + 6d − d 2 + 9 d 2 − 3d = d 2 + 6d + 9

−3d = 6d + 9 −9d = 9

d = -1

48. y

y + 2− 1y= 1

y

(y + 2)⋅ yy− 1y⋅ (y + 2)(y + 2)

= 11⋅ y(y + 2)y(y + 2)

y2

y(y + 2)− 1(y + 2)y(y + 2)

= y(y + 2)y(y + 2)

y2 − (y + 2) = y(y + 2)

y2 − y − 2 = y2 + 2y

−y − 2 = 2y

−2 = 3y

− 23= y