algebra i honors – quarter 4 exam review...quarter 4 exam review answer key 10-1 pythagorean...
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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
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