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Graphing Review
1. Using the table of values provided, graph the following line. Be sure to list at least 3 points.
π¦ =3
4π₯ + 2
x y
2. Using the table of values provided, graph the following line. Be sure to list at least 3 points.
2π₯ + 3π¦ = 6
x y
3. Given π¦ = β3π₯ + 1
a. create a table
b. Find the y-intercept
c. Find the slope
d. Graph (make sure to
label everything!)
3. There exists a line B with slope 3
4. Line C is parallel to line B and has a y-intercept of (0, 5). What is the
equation of the line? (2 marks)
4. The slope between the points ( 3,2 ) and ( -6,y ) is 2
3 . Calculate the value of y.
5. If the point ( 5,3) lies on the line y = mx + 2 find the slope.
6. The slope between the points ( -4, 1 ) and ( x, 4 ) is 3
4. Calculate the value of x.
7. Which one of the graphs matches the equation π₯ β 2π¦ = β4 (2 marks)
8. Which one of the following graph matches the equation π¦ = β2
3π₯ + 2
(2 marks)
a.
b.
c.
d.
9. Given the line A with slope 2
7 determine the equations for the perpendicular line which goes through the
point (14, 3). (2 marks)
Exponents Review
1. Evaluate the following (evaluate means find the answer!)
a. (-2)4
b. β(2)4 c. -2
4
d. (12+12)Γ82Γ20Γ·4 e. 5
-2
f. (1
4)
β3
g. (-2)0 h. -20
i. β(2)0
j. 32+3
-2 k. (22
)2Γ2
4Γ·(-2)
4
2. Simplify to positive exponential form
a. (-c)7(-c)
b. 6y-5
(-3y7)2 c. 5(3)
8Γ6(3)
4
d. (β36π2π2)
2
(12π2π2)3
e. 8(-7)4Γ4(-7)
11
f. (1
5)
β2
β 52
g. (3π 2)2 Γ (π 2
π‘)
β3
h. 3a+1
Γ 3-2a+1
i. 1
4Γ (2β2)β3
3. Use <, =, > to complete the following:
a. 22____2
-2
b.
5β2
23 _____ (2-3
52)-1
Answer Key:
1.
a. 16 b. -16 c. -16
d. 7680 e. 1/25 f. 64
g. 1 h. -1 i. -1
j. 82/9 or 91
9 k. 16 l.
2.
a. C8 b. 54y
9 c. 30(3)
12or 10(3)
13
d. 3
4m
-2n
-2 e. -32(7)
15 f. 0
g. 9π‘3
π 2 h. 3-a+2
i. 24
3.
a. >
b. <
Polynomials Review
1) Simplify
a. 4x + 3z β 8y β 5x +8z β 4y
b. 4x2 +8x β 7xy + 14x2y β 8xy2 + 17yx2 β 7x2 + 14xy
c. 3xy β 2yz + 5xz β 6yz + 4xy β 10 xz
d. yyyy6
1
4
1
3
2
2
1 22
e. 1.8s2t β 7.2st + 0.8st2 β 0.7s2t + 4.2st2 β 4.8st + st2
2) Evaluate the expressions for x = 3 y = 2 and z = -1 (simplify first!)
-3xy + 5yz β 2xz + 6xy + yz β 5xz
3) Write an expression for the perimeter of the figure
4) Simplify
a. (4x + 3) + (2x β 5)
b. (1 β 3t) β (-2 β 5t)
c. (3x + 2) β (x + 4) + (x -7)
d. (17n2 β 17n β 5) + (4n2 -3n +14) β (n2 + 13n + 23)
e. 4a2b β 5(a β b2) + a(a + 5)
f. a(a + b2) β 2(a2 β b2) +ab2
5) Expand
a. -4xy(x2 β y2)
b. 2xy2(y β 2x2y + 3xy) 6) Expand and simplify
a. (x + 3)(x β 4)
b. (5x + 3y)(4x β y)
c.
xx
4
133
2
1
d. (3x β 2)2
e.
2
3
1
x
f. (x + 4)(x2 -3x + 5)
g. 5(a + 1)(a β 1)
h. (x + 1)(x β 1)(x2 + 1)
i. (x β 5)3
q
r
q
p
y
x
7) Expand and simplify
a. 4x(x2 β 3x) β 4(3x2 β 2x)
b.
3
6
1
4
1
3
2
4
18
3
2xxxx
8) Find the area of the figure
9) Divide each polynomial
a. 33
25
9
54
ba
ba
b. 32
445362
16
483216
nm
nmnmnm
10) Expand and simplify
a. 3x(x2 β 3x + 3) β[4x(x2 β 3x + 4) β(x + 2)]
b. (x+4)(x β 2) β (x β 3)(x + 3)
c. (3xy +2x2y)(3x2 β 4xy + 3y2) - 4x[3x(x + 2y)- 2x(x β 3y)]
11) Calculate
a.
341
16
15
15
8
3
5
12) Simplify
b.
75
3322
4
422
yx
yxyx
Answers: 1) a. - x + 11z β 12y b. 31x2y β 8xy2 - 3x2 + 7xy + 8x c. 7xy β 8yz β 5xz d. βy2/4 + y/2 1.1s2t
β 12st + 6st2 2) 27 3) 2p + 4q + 2r 4) a. 6x β 2 b. 3 + 2t c. 3x β 9 d. 20n2 β 33n β 14 e.
4a2b + 5b2 + a2 f. βa2 2ab2 + 2b2 5) a. -4x3y + 4xy3 b. 2xy3 β 4x3y3 + 6x2y3 6) a. x2 β x β 12
b. 20x2 + 7xy β 3y2 c. βx2/8 +3x/4 + 9 d. 9x2 β 12x + 4 e. x2 β 2x/3 + 1/9 f. x3 + x2 β 7x + 20
g. 5a2 β 5 h. x4 β 1 i) x3 β 15x2 β 75x - 125 7) a. 4x3 β 24x2 + 8x b. 73x2/24 - 43x/36 8) 2xy β y2
9) a. -6a2/b b. βn3 + 2mn2 β 3m2n 10) a. -x3 +3x2 β 6x + 2 b. 2x + 1 c. 3x3y β 12x2y2 + 9xy3 β
4x3 β 24x2y 11) 163/84 or 1 12) 128x8/y2
Factoring and rational expressions Review
1) Find the GCF of: 10s4t5, 5s5t4, 15s3t4
Factor:
2) 12ab + 28ac + 36ad
3) x2 - x - 72
4) x3 +9x2 + 8x
5) 4x2 β 4y2
6) m4 - 16
7) 2a2 + 10a - 28
8) 10x + 24y
9) y2 β 15y + 56
10) 2x2 - x - 3
11) β15 + 8x β x2
12) 9 + 25x β 6x2 13)
16
1
9
4 2 x
14) 5xy2 β7x2y
15) 4x2 β 2x - 12
16) x4 β x2
17) 25m2 β (n + 2p)2
Expand and simplify
18) (3x β 4)(2x β 5)
19) Subtract the product of (4x2 β 2)(-3x2 + 2x + 5) from (x4 + 8x3 β x2 β 4x β 5)
20) (-5x β 3y)2
Multiply or divide the following rational expressions
21) 3
3
6
92
x
x
22) 49
882
10
3552
2
m
mmm
23) 4
483
16
16 2
2
2
x
x
x
x
Add or subtract:
24) 2
5
3
3
aa 25)
2
3
4
72
xx
x 26)
107
1
3011
2
128
3222
xx
x
xx
x
xx
x
27) What polynomial must be subtracted from (7x3 + 2x2 β 8x + 7) to give (3x2 β 4x β 9)
Bonus
28) Solve for x: 7
6
232
62
2
xx
xx
Bonus
29) 36
37
17
1
52
5
xx
30) Do the following division. State the result in the form: dividend = divisor x quotient + remainder
(4x3 β 4x + 3) (2x β 1)
Answers
1) 5s3t4 2) 4a(3b + 7c +9d) 3) (x + 8)(x β 9) 4) x(x + 1)(x + 8) 5) 4(x β y)(x + y)
6) (m β 2)(m + 2)(m2 + 4) 7) 2(a β 2)(a + 7) 8) 2(5x + 12y) 9) (y β 7)(y β 8)
10) (x + 1)(2x β 3) 11) -1(x β 3)(x β 5) 12) -1(3x + 1)(2x β 9) 13) (2/3x β ΒΌ)(2/3x + ΒΌ)
14) xy(5y β 7x) 15) 2(2x + 3)(x β 2) 16) x2(x + 1)(x β 1) 17) (5m β n β 2p)(5m + n +2p)
18) 6x2 β 23x + 20 19) 13x
4 β 27x
2 + 5 20) 25x
2 + 30xy + 9y
2 21)
2
3x 22)
7
22
m
m
23) 43
1
x 24)
2
5
a
a 25)
22
610
xx
x 26)
526
25193 2
xxx
xx 27) 7x3 β x2 β 4x + 16
28) x = 3 29) 5, -1/14 30) (2x β1)(2x2 + x β 3/2) + 3/2
Rational Expressions Review:
1. Simplify the following:
a. β36π¦4
28π¦3 b. 16π₯2
28π₯ c.
39π
13π4
d. 2π₯β2
π₯β1 e.
45
10π₯β10 f.
π₯β2
3π₯2β12
g. π₯β5
π₯2β10π₯+25 h.
π₯2β7π₯β30
π₯2β4π₯β60 i.
π₯2β11π₯+18
π₯2+2π₯β8
j. 9π₯+81
π₯3+8π₯2β9π₯
2 Simplify the following:
a. 2π
3Γ
3
2π2 b. 39π₯2
11Γ
22
13π₯ c.
6(π₯+1)
2Γ
4
π₯+1
d. (2π₯+3)
6Γ
2
6π₯2+9π₯ e.
7π₯2(π₯+4)
(π₯β3)(π₯+4)Γ
π₯β3
(π₯+8)(π₯+6) f.
1
π₯+10Γ
10π₯+30
π₯+3
g. π₯+7
7π₯+35Γ
π₯2β3π₯β40
π₯β8 h.
8π₯β56
8π₯+48Γ
π₯2+9π₯+18
8π₯2+24π₯
3. Simplify the following:
a. 10π
9Γ·
12π2
15 b.
7π₯2
7π₯3+56π₯2 Γ·2
π₯2+7π₯β8 c.
π₯2+10π₯+16
π₯2+6π₯+8Γ·
1
π₯+4
d. 10π₯2+42π₯+36
6π₯2β2π₯β60Γ·
40π₯+48
3π₯2β13π₯+10 e.
8
4π₯2β16Γ·
1
π₯β2
4. Simplify the following:
a. π’βπ£
8π£+
6π’β3π£
8π£ b.
5
3πβ6+
5π+1
3πβ6 c.
6
π₯+1β
5
4
d. π₯+1
7β
2π₯β3
7 e.
3
π₯+7+
4
π₯+7 f.
2π₯
π₯β1β
3π₯β4
2π₯+3
g. 7π
π+1+
8
πβ3 h.
3
8+
7
π₯β3 i.
5π+5
5π2+35πβ40+
7
π+8
j. 4πβ5
6π2+30π+
πβ1
6π k.
π₯+2
3π₯2+7π₯β20β
π₯+1
9π₯2β25 l.
π₯β4
π₯2βπ₯β20+
π₯+1
2π₯2+9π₯+4
KEY:
1a. -9y/7 b. 4x/7 c. 3/n3 d. 2 e. 9
2(π₯β1) f.
1
3(π₯+2) g.
1
π₯β5
h. π₯+3
π₯+6 i.
π₯β9
π₯+4
2a. 1/h b. 6x c. 12 d. 1/9x e. 7π₯2
(π₯+8)(π₯+6) f.
10
π₯+10 g.
π₯+7
7
h. π₯β8
8π₯
3a. 50/3n
b. π₯β1
2 c. x+8 d.
π₯β1
8 e.
2
π₯+2
4a. 7π’β4π£
8π£ b.
5π+6
3πβ6 c.
19β5π₯
4(π₯+1) d.
βπ₯+4
7 e.
7
π₯+7 f.
π₯2βπ₯β4
(π₯β1)(2π₯+3) g.
7π2β13π+8
(π+1)(πβ3)
or (7πβ8)(π₯β1)
(π+1)(πβ3)
h. 3π₯+47
8(π₯β3) i.
2(4πβ3)
(π+8)(πβ1) j.
π2+8πβ10
6π(π+5) k.
2(π₯2+3π₯+3)
(π₯+4)(3π₯β5)(3π₯+5) l.
3π₯2β11π₯β9
(π₯β5)(π₯+4)(2π₯+1)
Radicals Review:
1. Approximate the following:
a. β54
b. β106
2. Convert the following to mixed radicals in simplest form:
a. β125π b. 2β45π5
c. ββ625π3π43 d. β405π₯5π¦π§74
3. Simplify the following:
a. β2 Γ 6β6 b. 5β28 Γ 2β21
c. β3 Γ β21 d. β283
Γ β63
e. β2 Γ β3
4. Simplify the following:
a. 3β2 + β2 b. β27 + 2β12
c. 3β5 + 4β5 β β5 d. 5πβ18π2π + πβ8π3
e. 7π₯β8π₯π¦2 β 4π¦β32π₯3 + 9β8π₯2π¦3 f. 3β27π2 β 2β108π2 β β48π2
5. Simplify the following:
a. 3β3(4 β 3β5 ) b. 4β15(β3β6 + 5)
c. β15(2β10 β 4β6) d. βπ₯(3βπ₯ + 5)
e. (2 + β3)(3 - 2β5) f. (-7+β3π₯)(4 + β3π₯)
g. (2 - β5)(2 + β5) h. (β3 + β5π₯)(β3 + β5π₯)
6. Simplify the following:
a. β8
β7 b.
β7
8β7 c.
β21
β15
d. β14
3β35 e.
2β5
8β3
7. Determine the unknown side length:
a. b.
8. Convert the following to entire radicals
a. 3β2 b. 5πβ2π c. β12π₯3β3π₯
KEY
1a 7.3 b. 10.3
2a. 5β5π b. 6π2β5π c. β5ππ β5π3
d. 3xyzβ5π₯π¦π§34
3.a 12β3 b. 140β3 c. 3β7 d. 2β21 e. β6
4a. 4β2 b. 7β3 c. 6β5 d. 47abβ2π e. -2xyβ2π₯ + 18π₯π¦β2π¦
f. -7cβ3
5a. 12β3-9β15 b. -36β10 + 20β5 c. 10β6 β 12β10 d. 3x+5β3 e. 6+4β5 + 3β3 β 2β15
f. β21 β 3β3π₯ + 3 g. -1
6a. 2β14
7 b.
1
8 f.
β35
5 g.
β10
15 h.
β15
12
7a. π = 10 b. π₯ = 4
8a. β18 b. β50π3 c. ββ432π₯7
Solving Linear Equations Unit Review:
Part 1 β Type 1 to 4 Equations:
1. 4b β 6 = 14 2. β12m β 10 = 2 3. 3b + 5 = β13 4. 4w β 7 + 3w = 35
5. 12g β 72 = 4g + 8 6. β12 = 3 β 2m β m 7. 6 = β3(d + 2) 8. 75 = 3(β6n β 5)
9. β3(1 + 6r) = 14 β r 10. β4(3m + 2) β 5 = 2m β 3(m β 3)
Part 2 β Removing Fractions and Decimals:
1. 0.25(8y + 4) β 17 = β0.5(4y β 8) 2. h
3+
2
9=
5
6
3. m+3
2= β5 4.
2(nβ2)
3=
1
6 5.
4c
5β
4+5c
3= 3 6.
1
8(m + 5) =
7
4
7. 1
2(t β 1) β
2
3(t + 1) = 1 8.
2
dβ7=
3
dβ2
9. 6
p=
1
pβ5β
p+4
p2β5p 10.
1
b2β7b+10+
1
bβ2=
2
b2β7b+10
Part 3 β Literal Equations and Transforming Formula:
1. Make x the subject (solve for x) for the following:
a. π + π = ππ₯ + π b. π =5
9(π₯ β 32)
c. π₯
2π+
π₯
3π= π d. π(π₯ β π) + π(π₯ + π) = π
2. Make v the subject (solve for v):
a. π = ππ‘2 + π£π‘
Part 4 β Quadratic Equations:
Solve for m:
1. π2 β 6π + 8 = 0 2. π£2 + 4π£ β 21 = 0 3. 4π2 β 12π = 0
4. 9π2 β 16 = 0 5. (π₯ β 8)(π₯ + 1) = 10 6. (π₯ β 2)(5π₯ β 4) + 1 = 0
7. (2π₯ + 3)(3π₯ β 2) + 2π₯ + 3 = 0
8. (5π‘ β 1)(3π‘ β 2) = 22 9. π2 + 5π β 35 = 3π
11. 5π2 β 44π + 120 = β30 + 11π
Part 5 β Word Problems:
1. A certain number added to its square is 30. Find the number.
2. The product of two consecutive odd integers is 99. Find the integers.
3. The length of a rectangle exceeds its width by 4 inches. Find the dimensions of the rectangle it its
area is 96 square inches.
4. The ages of three family children can be expressed as consecutive integers. The square of the age of
the youngest child is 4 more than eight times the age of the oldest child. Find the ages of the three
children.
5. The height of a triangle is 5 less than its base. The area of the triangle is 42 square inches. Find its
base and height.
Answer Key:
Part 1:
1) B=5 2) m=-1 3) b=- 6 4) w=6 5) g=10 6) m=5
7) d= -4 8) n= -5 9) r= -1 10) M= -2
Part 2:
1) Y= 5 2) h = 11/6 3) m= -13 4) n = 9/4 5) c = -5 6) m = 9
7) t= -8 8) d=17 9) p = 4 1/3 10) B =6
Part 3:
1.
a. π₯ =π+πβπ
π
b. π₯ =9
5π + 32
c. π₯ =6πππ
3π+2π
d. π₯ =π+ππβππ
π+π
2. a. π£ =π
π‘β ππ‘
Part 4:
1) m= 2, 4 2) v= 3, -7 3) m = 0, 3 4) p = 4/3, -4/3 5) x= -2, 9
6) x = 9/5, 1 7) π₯ =1
3, β
3
2 8)π‘ =
5
3, β
4
5 9) π = 5, β7 10) π = 5, 6
Part 5:
1) n = 5, -6 2) The #βs are 9 and 11 3) w=8 L=12 4) The ages are 10, 11 and 12
5) base= 12, height = 7
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