Review Problems for Basic Algebra I Students
Note: It is very important that you practice and master the following types of problems in order to be successful in this course.
Problems similar to these are presented in the computer homework under “Review Exercises”. Once you have mastered the problems
on this sheet, go to the computer program and complete the review on line to be graded.
1.
2.
3.
4.
5.
Review Problem
16
Simplify:
18
Reference section in text
0.1
Answer
8
9
Combine:
3
5
+
8
6
0.2
29
24
Combine:
5 2
7 9
0.2
31
63
Multiply:
7
8
x
12
28
0.3
1
6
0.3
8
7
Divide:
6
3
÷
14
8
6.
Add: 1.6 + 3.24 + 9.8
0.4
14.64
7.
Multiply: 7.21 x 4.2
0.4
30.282
8.
Multiply: 4.23 x 0.025
0.4
0.10575
9.
Write as a percent: 0.073
0.5
7.3%
10.
Write as a decimal: 196.5%
0.5
1.965
A
Review Problems for Basic Algebra II Students
Note: It is very important that you practice and master the following types of problems in order to be successful in this course.
Problems similar to these are presented in the computer homework under “Review Exercises”. Once you have mastered the problems
on this sheet, go to the MyMathLab and complete the review on line to be graded.
1.
Review Problem
Combine: 7 + (- 6) – 3
Reference section in text
1.2
Answer
-2
2.
Combine: - 1(- 2)(- 3)( 4)
1.3
- 24
3.
Combine: (-5)4
1.4
625
4.
Multiply: - 52
1.4
- 25
5.
Evaluate: 3(5 – 7)2 – 6(3)
1.5
-6
6.
Simplify: 5(2a – b) – 3(5b – 6a)
1.7
28a – 20b
7.
Evaluate: x2 – 3x for x = - 2
1.8
10
8.
Solve for x: 4x – 11 = 13
2.3
x=6
9.
Translate into an algebraic expression:
three more than half of a number
2.5
10.
Explain how you would locate the
point (4, -3) on graph paper.
3.1
3+
1
x
2
Count from the origin 4 squares to
the right. From that location count 3
squares down. Place a dot at this
final location.
B
Inequality Symbols
Place the correct symbol, < or >, between the two numbers.
1)
2

4
2)
6

5
3)
-1

-3
4)
-5

-2
5)
- 13

7
6)
-4

10
7)
7

-6
8)
3
 -5
9)
-8

-5
10) -2

-7
11) - 5
-9
12)
-10

-7
13) 4

-4
14)
7

15)
7

-6
16) 9

-7
17)
-12
19) 30

27
20)
22) -34

47
23)
25) -37
28) -90
0
 14
18) -10

3
33

16
21) -24

42
19

-31
24)
43

-36
 -29
26) -41

-27
27)
53

-71
 70
29)
 -64
30)
91

-67
33)
84

73
53
31) -53
 -81
32) -88

-67
34) 67

35)

-37
59
48
36) -55
The Opposite of a Number

53
1
Find the opposite number.
1)
7
2) 11
3) -4
4)
-5
5) -18
6) 34
7)
-28
8) -77
9) 66
Evaluate.
10)
|3|
11) | -3 |
12) | 7 |
13)
| -5 |
14) | 4 |
15) | -4 |
16)
| -17 |
17) - | 4 |
18) | 15 |
19)
| -17 |
20) | -16 |
21) | -24 |
22)
- | 19 |
23) - | 21 |
24) - | -19 |
25)
- | -13 |
26) | -26 |
27) - | 22 |
28)
- | 31 |
29) - | -35 |
30) - | -33 |
31)
| 30 |
32) | 21 |
33) | -39 |
34)
| -28 |
35) - | 33 |
36) - | 43 |
2
Rules for Combining Signed Numbers - 1.1, 1.2
Rule 1: If the signs of the numbers to be combined are the same, then add the numbers and
keep the common sign as part of your answer.
Examples: + 6 + 3 = + 9;
+ 5 + 3 = + 8;
- 5 - 4 = - 9;
- 6 - 4 = - 10
7 + 4 = + 11 (Notice that the 7 has no sign, so we know it is a + 7)
Exercise A:
1) + 6 + 7
2) - 9 - 6
3) 12 + 5
4)
- 16 - 4
5) - 8 - 3
6) + 7 + 7
7)
17 + 6
8) - 5 - 5
9) + 10 + 5
10)
-8-4
11) 12 + 3
12) - 4 - 3
13)
+7+5
14) - 20 - 40
15) + 12 + 12
Rule 2: To combine numbers with different signs, subtract the numbers and take the sign of
the larger number for your answer.
Examples: - 5 + 3 = - 2 (The answer is negative since 5 is greater than 3, and 5 is
negative.)
- 7 + 8 = + 1 (The answer is positive since 8 is greater than 7, and 8 is
positive.)
- 4 + 9 = + 5 (The answer is positive since 9 is greater than 4, and 9 is
positive.)
+10 - 13 = - 3 (The answer is negative since 13 is greater than 10, and
13 is negative.)
Exercise B:
1) + 5 - 4
2) -3 + 8
3) 17 - 7
4)
+12 - 6
5) + 15 - 15
6) - 7 + 7
7)
+4-7
8) -40 + 10
9) 44 - 11
10)
- 32 + 2
11) 12 + 3
12) - 3 + 15
13)
+ 5 - 10
14) - 9 + 9
15) 7 - 9
Exercise C:
1) + 6 + 5
2) - 4 + 4
3) - 8 + 8
4)
+ 15 - 25
5) + 19 - 19
6) 9 + 4
7)
- 15 - 4
8) 17 + 7
9) 0 - 17
10)
21 - 0
11) - 9 - 5
12) + 16 + 4
13)
+ 5 + 12
14) -18 + 6
15) 24 + 2
16)
12 - 6
17) + 20 - 15
18) - 6 - 12
19)
+ 16 + 10
20) - 7 + 7
3
Combining Signed Numbers - 1.1, 1.2
(a) When the signs of numbers are the same or alike, add the numbers and keep the same
sign.
Examples: a. 3 + 5= + 8 b. - 2 - 12 = - 14 c. + 45 + 8 = 53 d. - 17 - 4= - 21
(b) When the signs of the numbers are different or unlike, subtract the smallest number from
the largest, and then take the sign of the largest number.
Examples: a. - 28 + 12 = - 16 b. + 9 - 45 = -36 c. 12 + 4 = 16
d. + 8 - 2 = 6
Add the following problems:
1) + 2 + 10 =
2) - 2 - 2 =
3) - 4 - 10 =
4) 2 + 5 =
5) - 15 - 13 =
6) - 2 - 10=
7) + 6 - 15 =
8) - 15 + 17 =
9) + 4 - 12 =
10) - 3 + 17 =
11) + 6 - 9 =
12) - 5 + 9 =
13) - 35 - 15 =
14) - 1 - 4 + 8 =
15) 1 + 2 + 3 =
16) - 2 - 4 - 6 =
17) - 4 - 10 =
18) + 34 + 38 =
19) - 21 + 5 =
20) + 3 - 5 =
21) + 13 - 12 =
22) - 19 + 9 =
23) + 11 - 10 =
24) - 6 + 6 =
25) + 3 + 21 =
26) + 8 - 23 =
27) - 13 - 13 =
28) + 4 - 29 =
29) - 4 + 23 =
30) - 15 + 8 =
31) - 1 + 45 =
32) - 3 + 1 =
33) 15 + 5 =
34) 23 + 2 =
35) - 9 + 12 =
36) 7 + 13 =
37) - 34 + 2 =
38) 9 + 17 + 12 =
39) - 2 - 4 - 17 =
40) 3 + 6 =
41) - 5 + 5 + 1 =
42) - 8 + 3 + 8 =
43) + 8 - 12 =
44) - 16 - 12 =
45) - 2 + 11 =
46) + 90 - 90 =
47) - 13 + 11 =
48) - 1 - 6 - 6 =
49) + 45 - 51 =
50) - 13 - 5 =
51) + 10 - 14 =
52) - 22 + 3 =
53) - 33 + 7 =
54) + 5 - 6=
55) + 2 - 11 =
56) + 45 - 45 =
57) - 30 + 20 =
58) + 16 - 15 =
59) + 2 - 2 =
60) + 40 - 10 - 30 =
4
Double Signs - 1.2
Always change double signs to a single sign before combining with RULES 1 or 2 from the previous
worksheet.
For example:
a) +5 + ( + 7) (Add the numbers, keep the common sign.) = + 5 + 7 = + 12
b) 5 - (- 7) (Add the numbers, keep the common sign.) = + 5 + 7 = + 12
c) 5 - (+ 7) (Subtract the numbers, keep the sign of the larger number.) = 5 - 7 = - 2
d) 5 + (- 7) (Subtract the numbers, keep the sign of the larger number.) = 5 - 7 = - 2
Add or Subtract:
(A)
1) +4 + (+ 2) =
2) + 4 + (- 2) =
3) + 4 - (- 2) =
4) + 4 - (+ 2) =
5) - 6 + (+ 3) =
6) - 6 - (- 3) =
7) - 6 + (- 3)
8) - 6 - (+ 3) =
9) 8 - (- 10) =
10) 8 + (- 10) =
11) 8 - (+ 10) =
12) 8 + (+ 10) =
1) 5 + (+ 2) - (+ 6) =
2) - 9 - (+3 ) + 5 =
3) 17 - (+ 7) - ( - 5)
4) - 4 - (+ 5) - (- 8) =
5) 15 - (+ 16) - (- 1) =
6) 8 + 3 - (+ 6) =
7) 20 + (- 23) - 5 =
8) - 4 + (+ 6) - (- 6) =
9) 30 - (+ 15) - (-5) =
10) 14 + 10 + (- 7) =
11) - 12 + 4 - 7 + (- 2) =
12) 3 + (- 4) - (- 3) +5 =
(B)
5
Review of Combining Signed Numbers - 1.2
Add or Subtract:
1) 3 + (- 6)
2) + 12 + 8
3) -11 + (- 16)
4) - 11 - (- 15) + 40
5) 3 + (- 9) + (- 7)
6) + 4 + (- 8) + (- 14)
7) - 9 + (3)
8) 1 + (- 2) + (- 3)
9) 11 +(- 20) + (- 30)
10) 12 + (- 20) + 2 + (- 7)
11) - 12 + 3 + 1 + (- 9)
12) - 21 + (- 10) + (- 6) + (- 15)
13) 15 - 6
14) 6 - 15
15) 13 - (- 14)
16) - 13 - (- 4)
17) - 9 - 6 - 5
18) 12 - 5 - 7
19) - 5 - (- 7) + (- 6)
20) 23 - 1 - (- 14)
21) 9 - 8 - (- 17)
22) - 8 - (- 26) - 41 + 16
23) - 14 - 20 - 12 - 15
24) 13 + (- 13) - 13 - (- 13)
25) - 71 + 64 - (- 23) + 1
26) 55 - 33 - 66 - 11
6
Multiplication - 1.3
(a) The product is positive if the two signs are the same, either both positive or both
negative. Examples: -5(-4) = +20; (+4)(+2) = +8
(b) The product is negative if the two signs are different, one positive and one
negative. Examples: (-8)(4) = -32; +6(-5) = -30
(c) When multiplying more than two terms, the product is positive if there is an even
number of signs and the product is negative if there is an odd number of negative
signs.
Examples: (-2)(-1)(8) = +16
(-1)(3)(+2) = -6
-2(-10)(-5) = -100
-4(2)(-2)(+2) = 32
1) (-3)(5) =
2) -8(-3) =
3. (-1)(-1) =
4) (+9)(7) =
5) +5(-10) =
6) (-6)(-2) =
7) (-7)(-8) =
8) -3(-9)=
9) -8(4) =
10) (-2)(-3) =
11) +6(+7) =
12) -8(+8) =
13) -3(-3) =
14) (-7)(-5)=
15) (-5)(3)=
16) (-2)(-2)(4)=
17) -3(6)(-6)=
18) -8(-7)(-1) =
19) (5)(5)(+3) =
20) (3)(-4)(+8) =
21) (1)(-1)(-1) =
22) (-2)(+2)(2) =
23) -6(-3)(-2) =
24) 3(+2)(-6) =
25) 5(-2)(-2) =
26) (1)(-2)(-1) =
27) (7)(3)(-2)=
28) (+5)(-2)(3)=
29) (+4)(+2)(3)=
30) (-8)(8)(4) =
31) -7(-6)(-2) =
32) -2(+5)(-3)=
33) -4(+5)(-2)=
34) -3(-2)(-7) =
35) (-5)(-5)(5) =
36) +8(2)(-1/4) =
37) (3)(-3)(-1/3)
38) -4(-2)(3)(-1) =
39) (+3)(7)(1/7) =
40) -5(1)(-1)(-1) =
41) (-3)(-2)(-7)=
42) (-9)(5)(1/9) =
43) (2)(2)(2)(-1/2) =
44) (-4)(2)(3)(-3) =
45) -10(3)(2)(1/5) =
46) (5)(6)(3)(1/3) =
47) 5(-2)(5)(-2) =
48) (-2)(5/6)(4)(-3) =
Multiply:
7
Division - 1.3
The rules used in multiplication are also used in division.
(a) The quotient is positive if the two signs are the same, either both positive or both
negative. Examples:
;
;
(b) The quotient is negative if the two signs are different, one positive and one
negative. Examples: (+8) ÷ (-4) = -2;
;
Divide:
1)
6)
11)
2)
3)
4)
5)
7)
8)
9)
10)
12)
13)
14)
15)
16) +35 ÷ (-5) =
17) (-13) ÷ (-13) =
18) (+22) ÷ (-2) =
19) -16 ÷ (-4) =
20) (-51) ÷ (17) =
21) 90 ÷ (-15) =
22) (46) ÷ (+2) =
23) 38 ÷ (-2) =
24) (-75) ÷ -5 =
25) -24 ÷ (-6) =
26) -81 ÷ (9) =
27) (+32) ÷ (+8) =
28) (-45) ÷ (-9) =
29) +80 ÷ (-16) =
30) (+28) ÷ -14 =
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
41)
42)
43)
44)
45)
46)
EXPONENTS - 1.4
A. Raise each base to its given power:
8
1) 22 = _____
2) 32 = _____
3. 42 = _____
4) 52 = _____
5) 62 = _____
6) 72 = _____
7) 82 = _____
8) 92 = _____
9) 102 = _____
10) 112 = _____
11) 122 = _____
12) 12 = _____
13) 23 = _____
14) 33 = _____
15) 43 = _____
16) 53 = _____
17) 13 = _____
18) 03 = _____
19) 15 = _____
20) 112 = _____
B. Raise each base to its given power:
1) (-2)2 = _____
2) (-3)2 = _____
3. (-4)2 = _____
4) (-5)2 = _____
5) (-6)2 = _____
6) (-2)3 = _____
7) (-3)3 = _____
8) (-4)3 = _____
9) (-5)3 = _____
10) (-6)3 = _____
11) (-1)2 = _____
12) (-1)3 = _____
13) (-1)4 = _____
14) (-1)5 = _____
15) (-1)6 = _____
16) (+6)2 = _____
17) (-10)2 = _____
18) (-10)3 = _____
19) (+10)2 = _____ 20) (+10)3 = _____
C. Raise each base to its given power. Be careful. These are tricky.
1) -22 = _____
2) -62 = _____
3. -92 = _____
4) -23 = _____
5) -15 = _____
6) -43 = _____
D. Evaluate the following. (Remember, always simplify the
exponent in these problems before doing any addition, subtraction,
multiplication or division.
1) 33 - 52 = ____
2) 62 -30(-2) = ____
3. 12 -13 +14 = ____ 4) 23 +(-2)2 = ____
5) -7 - (-5)2 = ____ 6) 2(3)2 = ____
7) -6(2)2 = ____
8) (3)(4)2 = ____
9) -6(-2)3 = ____
11) -92 -72 = ____
12) 53 + 52 = ____
10) 5(-3)2 = ____
13) (-4)2 +(+3)3 = ____
14) 15 +(-2)3 = ____
15) -82 +23 = ____
9
Order of Operations - 1.5
Simplify using the order of operations (PEMDAS).
1)
10 - (-2)3 - (-1)
2)
3(5 - 1) - (-2)2
3)
5 - 3[9 - ( - 3)]
4)
-32 + 2[8 ÷ (1 + 3)]
5)
3[15 - (8 - 5)] ÷ 6
6)
5[20 - (9 - 4)] ÷ 25
7)
6[14 - (11 - 9)] ÷ 32
42
8) 32 ÷
- (-6)
7-5
9)
11 +
19 - 3
-7
32 - 1
10)
(5 - 2)3 - 4 - 2 · 3
10
EQUATIONS - 2.1
Solve the equations:
1) a + 5 = 7
2) y + 1 = 6
3) k + 4 = 1
4) 5 + a = 12
5) 3 + y = -7
6) 2 + z = 0
7) 9 = 8 + c
8) 1 = 1 + x
9) y + 4 = -4
10) -12 = k + 8
11) y - 5 = 1
12) x - 10 = 6
13) k - 1= - 4
14) m - 7 = 0
15) 12 = y - 6
16) 4 = x - 4
17) a - 7= - 7
18) 3=n-3
19) 0 = n - 1
20) -6 = y - 8
21) 6x= 18
22) 3y = 21
23) 4a = - 8
24) 30 = 10k
25) 1 = 4x
26) 5x = 0
27) -48 = 8m
28) -14y = -7
29) -x = -2
30) 11 = -k
31) 2x + 3 = 13
32) 7 + 12k = -53
33) 7x -2 = 33
34) 5a + 7 = 12
35) 25 = 6x + 1
36) 10k -7 = 23
37) 6m + 2 = -16
38) 3 = 4z + 3
39) -4 + 2a = 12
40) 10 + 3y = 25
11
Two-Step Equations - 2.2
Two Step Equations With Four Terms: The proper procedure is to move the variables (x’s) to
one side of the equation and to move all the constants/numbers to the other side of the equation.
Examples
a. -6x +4 = -8x +10
+8x -4
+8x -4
2x
=
6
x=3
b. 3x - 5 = 13x +15
-3x -15
-3x - 15
-20 = 10x
-2 = x
Solve the equations:
1) 6x + 6 = 8x + 2
2) 12x + 6 = 8x - 10
3) 5x + 8 = 8x - 1
4) 7x - 11= 14x + 10
5) 2x - 8 = 5x - 23
6) 10x + 4 = 8x - 8
7) 5x + 13 = 3x - 13
8) 9x + 11 = 6x + 14
9) 2x - 8 = 4x + 4
10) 16x - 2 = 14x - 34
11) 4x - 1 = 13x - 19
12) 8x + 11 = 7x - 17
13) 20x + 10 = 10x + 60
14) x - 8 = 5x + 4
15) x - 1 = 2x - 1
16) 4x - 9 = 3x + 9
17) 5x + 15 = 10x + 25
18) 2x - 14 = 19x + 3
19) 5x + 12 = 6x + 7
20) 7x - 6 = x + 9
21) 15x + 14 = 10x + 4
22) x - 12= 2x - 2
23) 6x - 5 = -4x + 10
24) 4x + 7 = 13x - 8
12
Multi-Term Equations - 2.2
Multi-Term Equations - If an equation has more than one of the same term on either side of the
equation, the like terms should be combined before solving the equation.
Example
2y + 8 - 14 =
2y - 6 =
- 2y +12
6 =
1 =
5y - 12 + 3y
8y - 12
-2y +12
6y
y
(On the left side of the equation, the +8 and
the -14 are combined first. On the right side
of the equation, the 5y and the 3y are
combined first.)
Solve:
1) 8x - 3 + 5 = 3x + 22 + 4x
2) 12x - 10 - 2x = 12 + 7x - 7
3) 4 + 8x + 12 = 4x - 20 - 2x
4) 11x + 6 + 3x = -19 + 3x - 8
5) - 4 - 6x + 5 = 3x + 12 + 2x
6) 16 + 9x - 6 = 4x + 8 + 3x
7) 5x - 34 - 7x = 21 + 4x + 11
8) 6x + 3x - 5 = 4x + 22 - 7
9) 13x - 19 - 3 = 3x + 5x + 18
10) 22 + 3 - 7x = x + x - 11
11) - 2x + 9x - 4x = 24 - 13 + 8
12) 8 - 2x + 7x = 2x + 16 - 7
13) 7x - 25 - 2x = 15 - 3x + 16
14) 4x - 2 + 3x = 12 + 3 - 7x
15) 13 + 8x + 14 = 52 + 9x + 3x
Equations With Parentheses - 2.3
13
Equations With Parentheses - The proper procedure is remove all parentheses on both sides of
the equation and then to combine like terms before solving.
Example:
7 + 2(x - 4) = 6x - (5x + 10)
7 + 2x - 8 = 6x - 5x - 10
2x - 1 = x - 10
- x +1 = -x + 1
x= -9
(parentheses removed)
(terms combined)
Solve:
1) 8 + 3(x + 2) = 4x - (2x + 5)
2) 2 +3(x + 6) = 11 - (5x + 15)
3) 3(x + 4) = 5 - (x - 11)
4) 8(x + 12) = 3(x - 18)
5) x - 4(x - 7) = 2(3x - 13)
6) 3(x - 3) + 3 = 3x - (3x - 3)
7) 7(3x + 1) = 3(2x + 8)
8) -1 + 8(8 - x) = 4 - (4 - x)
9) 9(2x + 3) = 3(x - 6)
10) 3 - 6(x - 3) = 4x + 3(x - 8)
11) 3x - 2(x - 7) = 3(2x - 3) - 7
12) 6x + 7(x - 2) = - 2(x - 5) - 11
13) 5x + 3(2x + 3) = 12 - (2x - 5)
14) 31 + 5(5x + 3) = 13 + 3(3x + 9)
15) 11(x - 2) = 22 - 2(7x - 3)
14
Supplementary Equations - 2.4
Solve:
2) 3x + 5 = x + 7
1)
x+
=5
3)
(x + 6) = x + 4
4) x =
5) 4x - 3 = x - 9
7)
x+
9) 2x -
8) 3x - 1 = 2(x - 5)
10) 3x + 2 = 5x - 8
x
11) 3(x + 4) + 1 = 9 - x
13) x - 7 = 4x + 5
21) 5x +
18)
x+1
=5-
x
=
x+2
(2x + 4) =
16) 4x -
17) x - 6 = 5x -14
x-5=
12) 2x -
14)
15) 2x +3 = 3x + 5
19)
x
6) 3 - x = 2(1 - x)
=
=
-
x+
=
=
(x - 5)
x+3
+
20) 3x + 7 = 2x +
x
(2x +1)
22) 3x + 7 = 5x - 4
Supplementary Equations (Cont.)
Solve:
23)
24) 3x + 2(x - 5) = 7 - (x + 3)
15
25) 3 - x =
(7 + 2x)
26)
27) 5x - 3(x + 1) = 5
29)
x+6-
x=
31)
(x + 3) + 8 =
33)
x = 11 +
28) x +
x+9
x + 11
x-
=
x+4+
x
=
x+3
= 3x +
30) 7x + 5 = 2(x - 1) - 21
32) 5 + 6x - 3 = 2 + 4x
34) x +
35) 5x - (2x - 3) = 4(x + 9)
37) 6x -
x-
=
36)
(x + 6) =
38)
=
39)
(4 - x) - 6 = x + 6
40)
41)
(x + 6) + 5 = 2x + 28
42) 2x +
x = 28 -
=
-
(2x - 5)
(10 - x)
x
x+1
16
Literal Equations - 3.1
Literal Equations ~ Equations that contain more than one letter
Example: 2x + 3y = 12. Solve for x.
Example: 2x + 3y = 12. Solve for y.
2x = - 3y + 12
2x -3y
12
=
+
2
2
2
-3
x=
y+6
2
3y = - 2x + 12
3y
-2x 12
=
+
3
3
3
-2
y=
x+4
3
Solve for the indicated variable:
1) x + y = 12. Solve for x.
2) 3x + 2y = -12. Solve for y.
3) a - b = 5. Solve for b.
4) 2a + 3b = 9. Solve for a.
5) 6x - 6y = 6. Solve for x.
6) x - 2y = 10. Solve for x.
7)
x
+ y = 6. Solve for x.
2
9) a +
b
= 2. Solve for a.
4
8)
2
a - 6b = 9. Solve for a.
3
10) 2x - 4y = 5. Solve for y.
11) x + y = 12. Solve for y.
12) 3x + 2y = 12. Solve for x.
13) a - b = 5. Solve for a.
14) 2a + 3b = 9. Solve for b.
15) 6x - 6y = 6. Solve for y.
16) x - 2y = 10. Solve for y.
17)
x
+ y = 6. Solve for y.
2
19) a +
b
= 2. Solve for b.
4
18)
2
a - 6b = 9. Solve for b.
3
20) 2x - 4y = 5. Solve for x.
17
T
a
b
l
e
Equation: y = 3x - 2
x
0
1
2
y
Equation: y = -2x + 4
o
f
V
a
l
u
e
s
x
0
1
2
y
Equation:
x
0
3
6
y
18
T
a
b
l
e
Equation: y = x2 - 4
x
-3
-2
-1
0
1
2
3
y
Equation: y = x2 - x - 2
o
f
V
a
l
u
e
s
x
-3
-2
-1
0
1
2
3
y
Equation: y = x2 +2x - 6
x
-3
-2
-1
0
1
2
3
y
19
T
a
b
l
e
Equation:
x
y
Equation:
o
f
V
a
l
u
e
s
x
y
Equation:
x
y
20
The Equation of a Line
y = mx + b
Find the Equation of the line given the following information:
A
Information given
Given: m (slope) and
b (y‐intercept or (0, b))
What you will need
The answer is…
Nothing!
Easy!
Examples
Given: m = 3, b = 7
Y = 3x + 7
2
Y= x-4
3
2
Given: m = , (0, - 4)
3
Find the Equation of the line given the following information:
2
1
5
1. m = ‐2, b =
2. m = 5, (0, 9)
3. m = , b = 5
4. m = ‐7, b = ‐2
5. m = , (0, 0)
5
5
7
***********************************************************************************
B
Information given
Given: m (slope) and a point
(that is not the y intercept)
Examples
Given: m = 5, (2, 7)
2
Given: m = , (3, ‐ 2)
3
What you will need
The y‐intercept or b.
• Use the m that is given
• Use the point (x, y)
• Replace the x, m, and y
• y = mx + b
( ) = ( )( ) + b
• Solve for b
y = mx +b
( ) = ( )( ) + b
7 = (5)(2) + b
7 = 10 + b
‐3 = b
Y = mx + b
( ) = ( )( ) + b
2
‐2 =( )(3) + b
3
‐2 = 2 + b
‐4 = b
Find the Equation of the line given the following information:
2
1
6. m = ‐2, (3, 3)
7. m = 5, ( , 5)
8. m = , (‐5, 0)
5
5
The answer is…
Use the m given
Use the b you found
Discard the point (x, y)
Write the equation of the line!
Y = 5x ‐ 3
2
Y= x‐4
3
9. m = ‐7, (2, 4)
5
10. m = , (7, 2)
7
20A
C
Information given
Given: Two points
(x1, y1) (x2, y2)
What you will need:
y2 ‐ y1
1. The slope: m =
x2 ‐ x1
2. The y‐intercept or b.
The answer is…
Use the m you found
Use the b you found
Now…
• Find m
• Use one of the points (x, y)
• Replace the x, m, and y
• y= mx + b
( ) = ( )( ) + b
• Solve for b
OR
Use the
Point‐Slope Formula:
y – y1 = m (x – x1)
Examples
Given: (‐2, 5) and (4, ‐1)
1.Find the slope:
y2 ‐ y1
‐1 ‐ 5 ‐6
m=
=
=
= ‐1
x2 ‐ x1 4 ‐ (‐2) 6
2.Now, use only one of the
points.
y = mx +b
( ) = ( )( ) + b
5 = (‐1)(‐2) + b
5 = 2 +b
3 = b
Given: (3, 2) and (‐3, 6)
Using the Point‐Slope
Formula:
y – y1 = m (x – x1)
m=
y2 - y1
6 - 2 4 -2
=
= =
x2 - x1
-3 - 3 -6 3
Discard both points
Write the equation of the line!
Y=‐x+3
y =
y – y1 = m(x – x1)
‐2
y – (2) = (x – 3)
3
‐2
y–2= x+2
3
‐2
y = x+4
3
‐2
x+4
3
Find the Equation of the line given the following information:
11. (4, ‐3), (‐1, 7)
12. (‐1, ‐5), (‐4, 1) 13. (2, 14), (‐4, ‐4)
14. (‐2, ‐6), (1,0)
15. (3, ‐1), (4, ‐1)
*************************************************************************************
Answers:
2
5
2. y = 5x + 9
6. y = ‐2x + 9
7. y = 5x + 3
1
+5
5
1
8. y = x + 1
5
11. y = ‐2x + 5
12. y = ‐2x – 7
13. y = 3x + 8
1. y = ‐2x +
3. y =
4. y = ‐7x ‐ 2
5
5. y = x
7
9. y = ‐7x + 18
5
10. y = x ‐ 3
7
14. y = 2x – 2
15. y = ‐1
20B
Multiplication of Monomials - 5.1
Multiplication of Monomials by Monomials - Three steps: a) multiply the signs, b) multiply the
numerical coefficients, and c) add the exponents of the same bases.
Examples:
a) (2x3)(+4x2y4) = +8x5y4
b. (-2a5b3)(-10a2b3)(3b2) = -60a7b8
Multiply:
1) x3 • x3
2) b8 • b3
3) y8 • y2
4) 4x4(10x2)
5) (5y8)(5y3)
6) (-2a3)(7a4)
7) (7y8)(-5y9)
8) -8x6(3x9)
9) (3b4c4)(-2b2c5)
10) (-9m)(+2m)
11) (+8x3)(4x8)
12) (2x2y5)(-14xy)
13) (-8x3)(-6x7)
14) (4x9)( x2)
15) (-2y2z2)(-7y3z3)
16) (-5x7y)(3xy3)
17) (-4x3)(4x8)(4x6)
18) (2x5)(-2x5)(-11x3)
19) (-10c)(3c3)(-2c5)
20) (-10b4)(2b5)(- b3)
21) a9b9 • a2b6
22) (-x2y2)(-x5y5)
23) (3a)(-7a5)( a5)
24) (-10x2)(-7x6)( x5)
25) (a2b3)(-2bc)(-2a5c5)
26) (14mn)(2mn)(2mn)
27) (-2a5b)(6b2)(5a2)
28) (6x5)(-8xy2)
29) (-3x5y)(-2x2y4)
30) (-y5)(+3y5)
21
Multiplication of Monomials (Exponents outside Parentheses) - 5.1
Multiplication of Monomials with Exponents outside of Parentheses - The
exponent outside a parentheses indicates the power to which the parentheses must be
raised.
Examples a.
b. If there is no numerical coefficient, multiply the
2 4
(2a )
exponents inside the parentheses by the exponent
= (2a2) (2a2) (2a2) (2a2)
that is outside the parentheses. (a3b5)6 =x18y30
= 16a8
Multiply:
1) (3r2s)2
2) (5x3y3)3
3) (-4x3y2)3
4) (8x3y4)2
5) (6a3b5)2
6) (-2x4y4)4
7) (10x)3
8) (3a2)4
9) (2y5z)3
10) (-4ab2)3
11) (a3b3)3
12) (-5xy2)3
13) (x5y2)6
14) (-2ab8)2
15) (3a3)2
16) (x5)5
17) (z7)3
18) (a2)3
19) (m3n4)5
20) (xy2)2
21) (-p3q3)7
22) (-a5b6)5
23) (-v6)6
24) (b3c3)3
25) (abc3)5
26) (x5y2)8
27) (-a4b4)5
28) (-a4c4)7
29) (x3y3z2)4
30) (d2)15
31) (4x3)4
32) (f7)6
33) (7x2y3)3
34) (x2y2)5
35) (-5a5b4)3
36. (-m5n2)7
Multiplication of Monomials and Monomials
(Exponents outside the Parentheses) - 5.1
22
Examples
a. (3x2y2)2(2xy)3
= (3x2y2) (3x2y2) (2xy) (2xy)
(2xy)
= 72x7y7
b. (-2a5)3(a2b5)4
= (-2a5) (-2a5) (-2a5) (a2b5) (a2b5)
(a2b5) (a2b5)
= -8x23y20
Multiply:
1) (4xy)3(x3)2
2) (6x2y3)3(x2)4
3) (-2a2)2(a)3
4) (-3x2y) (xy3)3
5) (12x3)2(-2x2)3
6) (2x)4(-2x)
7) (4b2)2(2a2b3)2
8) (2x2)5(2x3y2)
9) (-8x)3(x4y3) 2
10) (mn2)4(-2)2
11) (b5)3(-5b3)2
12) (x2y)2(xy3)4
13) (x2y)4(-xy2)4
14) (xy3)(-3x3y2)
15) (4b3)3(-a3b2)
16) (2x5y7)(5xy4)2
17) (4x2y)3(xy2)
18) (2a3b4)2(8ab)2
19) (-3xy)3(xy7)4
20) (-4x2)2(x2)9
21) (-2mn)(-m4)4
22) (2pq)3(-4p2q2)2
23) (-3y4)3(x5y6)7
24) (r5s4)3(r5s4)3
23
Zero Exponents - 5.1
Zero exponents - any number, variable, or entire term raised to the zero power is equal to "1". The
only exception to this rule is "0" to the "0" power.
Examples:
a. xo = 1
b. xoy = 1y = y
d. a3boc = a3c
c.
Simplify:
1) ao =
2) yo =
3) ro =
4) (xz)o =
5) (ax)o =
6) (xyz)o =
7) xob =
8) roc =
9) xco =
10) a2bo =
11)
12)
=
13) a2box2 =
14) 3xyo =
15) -xyo =
16) (3a2) o =
17) 3(ab)o =
18) -3(c2d)o =
24
25
Division of Monomials - 5.1
Division of Monomials - If the largest exponent is in the numerator, the variable remains in the
numerator, but if the largest exponent is in the denominator, then the variable stays in the
denominator.
Examples
a.
b.
c.
d.
Simplify:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
Negative Exponents - 5.2.1
Negative Exponents - To change a negative exponent to a positive exponent, move the exponent
and its base from the numerator to the denominator. If the exponent is in the denominator, move it
to the numerator.
Examples
a.
b.
c.
Change all negative exponents to positive exponents and simplify.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
21)
22)
23)
24)
27)
28)
25)
26)
20)
26
Negative Exponents (Cont.) - 5.2.1
Write with a positive exponent. Then evaluate.
1)
2)
3)
4)
6)
7)
8)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
22)
23)
24)
25)
26)
27)
29)
30)
31)
5)
9)
Simplify.
21)
28)
32)
27
Addition and Subtraction of Polynomials - 5.3
Combining Polynomials - To add or subtract polynomials, combine the numerical coefficients of the
like terms. (Like terms are terms that have the same variables with the same exponents.)
Examples: a. (4x2 + 3x -2) + (2x2 - 5x -6)
b. (3a2 -5a + 2) - (4a2+ a + 2)
2
2
(4x + 2x ) + (3x - 5x) + (-2 -6)
3a2 - 5a + 2 - 4a = -a -2
6x2 - 2x - 8
(3a2 - 4a2) + (-5a - a) + (2 - 2)
-a2 - 6a
Simplify.
1) (x2 + 5x) + (-2x2 - 3x)
2) (y2 + 3y) + (-2y -5)
3) (x2 + 4x + 9) + (x2 -x -6)
4) (x3 -5x + 6) + (3x2 -x -6)
5) (4y3 + 2y2 -2) + (-3y3 -2y2 -1)
6) (x3 -3x) - (x2 - 7x)
7) (x2 -3x + 2) - (x2 + 6x + 7)
8) (3x3 + 6x + 3) - (-2x2 + 3x + 2)
9). (5y3 + 4y -1) - (y3 + y2 + 6)
10) (2x3 -5x + 6) - (x3 -x + 7)
11) (y3 -6xy + 2) + (y3 -6xy -7)
12) (x2 - 2xy) - (-2x2 + 3xy)
13) (2x2 + x -1) - (x2 + 6x -3)
14) (3x2 + 2x -2) + (x2 + 5x -6)
15) (3x3 -2x -6) - (2x2 +6x -1)
28
Distributive Property - 5.4
Simplify.
1) x(x + 1)
2) y(2 - y)
3) -x(x + 2)
4) -y(8 - y)
5) 2a(a - 1)
6) 3b(b + 5)
7) -2x2(x - 1)
8) -4y2 (y + 6)
9) -6y2(y2 - y)
10) -x2(2X2 - 3)
11) 2x(5x2 - 2x)
12) 3y(2y - y2)
13) (2x - 3)4x
14) (2y - 1)y
15) (2x - 3)x
16) (2x - 1)3x
17) -x2y(x - y2)
18) -xy2(2x - y)
19) x(x2 - 2x + 1)
20) x(x2 - 3x - 2)
21) y(-y2 + 4y - 3)
22) -y(y2 - 5y - 6)
23) -a(a2 - 6a - 1)
24) -b(2b2 + 3b - 6)
25) x2 (2x2 - 3x - 2)
26) y2 (-3y2 - 5y - 3)
27) x3(-x2 - 5x - 6)
28) y3(-2y2 - 3y - 4)
29) 2y2(-2y2 - 5y + 8)
30) 3x2(4x2 - 2x + 7)
31) 4x2(5x2 - x - 9)
32) 5y2(-y2 + 3y - 6)
33) xy(x2 - xy + y2)
34) ab(a2 - 3ab - 4b2)
35) xy(x2 - 2xy + 2y2)
36) ab(a2 + 5ab - 7b2)
29
Multiplying Binomials
Simplify.
1) (x + 1)(x + 4)
2) (y + 2)(y + 3)
3) (a - 2)(a + 5)
4) (b - 5)(b + 4)
5) (y + 2)(y - 7)
6) (x + 9)(x - 4)
7) (y - 6)(y - 2)
8) (a - 7)(a - 8)
9. (a - 2)(a - 8)
10) (x + 11)(x - 3)
11) (2x + 1)(x + 6)
12) (y + 1)(3y + 2)
13) (2x - 3)(x + 3)
14) (5x - 2)(x + 3)
15) (3x - 2)(x - 5)
16) (2x - 1)(3x - 5)
17) (2y - 9)(y + 1)
18) (4y - 7)(y + 2)
19) (3x + 4)(3x + 7)
20) (5a + 2)(6a + 1)
21) (6a - 13)(2a - 5)
22. (5a - 9)(2a - 7)
23) (3b + 11 )(5b - 4)
24) (3a + 10)(4a - 3)
25) (x + y)(x + 2y)
26) (2a + b)(a + 2b)
27) (2x - 3y)(x - y)
28) (a - 3b)(2a + 3b)
29) (4a - b)(2a + 5b)
30) (2x - y)(x + y)
31) (3x - 5y)(3x + 2y)
32) (5x + 2y)(6x + y)
30
Special Products - 5.5
Simplify.
1) (x + 1)(x - 1)
2) (x - 3)(x + 3)
3) (x + 5)(x - 5)
4) (x - 7)(x + 7)
5) (2x - 1)(2x + 1)
6) (3x - 1)(3x + 1)
7) (4x - 3)(4x + 3)
8) (x + 5)2
9) (y - 4)2
10) (3y - 1)2
11) (x - 1)2
12) (x - 3)2
13) (x + 7)2
14) (x + 9)2
15) (x - y)2
16) (2a - 5)2
17) (5x - 4)2
18) (3x - 7)2
19) (3a - 5)(3a + 5)
20) (6x + 5)(6x - 5)
21) (2x + 5)2
22) (9x - 2)2
23) (a - 2b)2
24) (x + 2y)2
25) (5x - 6)(5x + 6)
26) (b - 6a)(b + 6a)
27) (x + 5y)2
28) (2 - 7y) 2
29) (3 - 5y) 2
30) (3 - 5y)(3 + 5y)
31) (4x - 1)(4x + 1)
32) (2a + 3b)2
33) (x + 6y)2
31
Applications with Polynomials
Solve.
1. The length of a rectangle is 3x. The
width is 3x - 1. Find the area of the
rectangle in terms of the variable x.
2. The width of a rectangle is x - 2. The
length is 3x + 2. Find the area of the
rectangle in terms of the variable x.
3. The length of a rectangle is 3x + 1.
The width is 2x - 1. Find the area of the
rectangle in terms of the variable x.
4. The width of a rectangle is x + 7. The
length is 4x + 3. Find the area of the
rectangle in terms of the variable x.
5. The length of a side of a square is
x + 3. Use the equation A = s2 where s
is the length of a side of a square, to
find the area of the square in terms of
the variable x.
6. The length of a side of a square is x - 8.
Use the equation A = s2. where s is the
length of the side of a square, to find the
area of the square in terms of the variable
x.
7. The length of a side of a square is
2x + 1. Find the area of the square m
terms of the variable x.
8. The length of a side of a square is 3x - 4.
Find the area of the square in terms of the
variable x
9. The radius of a circle is x + 4. Use the
equation A = πr2 where r is the radius, to
find the area of the circle in terms of the
variable x.
10. The radius of a circle is x - 3. Use the
equation A = πr2, where r is the radius, to
find the area of the circle in terms of the
variable x.
11. The radius of a circle is x + 6. Find
the area of the circle in terms of the
variable x.
12. The radius of a circle is 2x + 1. Find the
area of the circle in terms of the variable x.
32
Dividing a Polynomial by a Monomial - 5.6
Simplify.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22.
23)
24)
25)
26)
27)
28)
33
Removing a Common Factor - 6.1
Factor.
1) 4a + 4
2) 6c - 6
3) 8 - 4a2
4) 9 + 21x2
5) 3x + 9
6. 10a2 + 15
7) 24a - 8
8) 24x + 12
9) 9x - 6
10) 16a2 - 8a
11) 12xy - 16y
12) 6b2 - 5b3
13) 20x3 - 24x2
14) 12a5 - 36a2
15) 24a3b4 - 18a2b2
16) 4a5b + 6ab4
17) a3b2 + a4b3
18) 25x2y2 - 15x3y
19) 3x2y - 5xy2
20) 8a3b2 - 12a2b3
21) x3y2 - x2y4
22) x3 - 5x2 + 7x
23) y3 - 6y2 - 8y
24) 4x2 - 16x + 20
25) 6y2 - 9y + 12
26) 3X2 - 9x + 18
27) b4 - 3b3 + 7b2
28) 4x2 - 8x3 + 12x4
29) 12y2 - 16y + 48
30) 5y4 + 10y3 - 35y
31) 4x4 + 12X3 - 28X2
32) 45a4b2 - 75a3b + 30ab4
33) 32x4y2 - 96x2y4 - 48x6y2
Factoring by Grouping - 6.1
34
Factor:
1) x(a+ b) + 3(a + b)
2) a(x - y) + 5(x - y)
3) x(b -1) - y(b - 1)
4) a(c - d) + b(c - d)
5) y(a - 1) - (a - 1)
6) a(y + 3) - (y + 3)
7) x(y - 2) - (y - 2)
8) 3x(y - 7) - (y - 7)
9) 2x(x - 5) - (x - 5)
10) 5y(x - 3) + (3 - x)
11) x(a - 2b) + y(a - 2b)
12) 4a(a + 1) - (a + 1)
13) a(x - 9) - (x - 9)
14) b( a - 5) -2(a - 5)
15) c (x - 3y) + d(x - 3y)
16) 3x(a + 1) + 4(a + 1)
17) a(x - 1) + 3(1 - x)
18) x(a - 4) + y(4 - a)
19) x(a - 3b) + y(3b - a)
20) x(a - 9) - (9 - a)
21) a(x - 7) + 3(7 - x)
22) 2x(x - 6) + (6 - x)
23) d(e - 5) + (5 - e)
24) 2x(a - 4) - y(4 - a)
25) m(a - 9) - n(9 - a)
26) 3m(n - 2) - (2 - n)
27) 2a(b - 1) - c(b - 1)
28) x(a - 2) - 3y(a - 2)
29) x(y - 5) + (5 - y)
30) 2(a - b) + c(b - a)
31) m(n - 1)+ 2(1 - n)
32) x(c - 7) - y(7 - c)
33) a(b - c) - 3(c - b)
35
Factoring by Grouping (Cont.)
Factor by grouping:
1) 2xy - 6x + 3y - 9
2) x2 + xy + 2x + 2y
3) ax + 5x + 6a + 30
4) 2x2 - 2xy + x - y
5) 6x2 + 2x + 6xy + 2y
6) 4a2 + 4ab + 3a + 3b
7) ax - 7a - 2x + 14
8) 2x2 + 2xy - 5x - 5y
9) ax + a - 2x - 2
10) 3xy + y - 9x - 3
11) 3a2 - a - 6ab + 2b
12) 2ax + x - 6a - 3
13) 2ax - 3x - 4a + 6
14) 5a2 - 15a - 3ax + 9x
15) xy - 3x - y2+ 3y
16) 7a2 - ay - 7ab + by
17) 6x2 - 4x - 3xy + 2y
18) 4a2 + 12a - ab - 3b
19) 2a2 + 3a - 8ax - 12x
20) 3x2 - 6x - xy + 2y
21) 8ax - 2a + 4xy – y
36
Factoring Trinomials with Coefficients of 1 - 6.3
Factor.
1) a2 - 2a - 35
2) a2 - 4a + 3
3) a2 + 3a - 10
4) a2 - 5a + 6
5) b2 - 7b + 10
6) b2 + 8b + 15
7) y2 + 5y - 66
8) x2 - 4x - 60
9) y2 - 7y + 10
10) y2 - 9y +18
11) x2 - 12x + 36
12) x2 - 4x - 96
13) a2 + 3a - 28
14) x2 + 10x +16
15) b2 - 11b - 180
16) x2 + 10x + 25
17) x2 - 14x + 49
18) b2 + 7b + 12
19) b2 + 10b + 16
20) x2- 9x - 36
21) x2 - 7x - 60
22) x2 + 10x - 56
23) x2 - 8x -128
24) x2 - 4x -77
25) b2 - 20b + 84
26) b2 - 21b + 108
27) b2 - 27b + 180
28) a2 + 16a + 63
29) x2 - 19x + 60
30) x2 - 25x + 84
37
38
39
Factoring Trinomials with Coefficients Greater than 1 - 6.4
Factor.
1) 2x2 - 5x + 2
2) 3x2 - 2x - 1
3) 2a2 + 7a+ 3
4) 3x2 + x - 2
5) 2b2 - 13b + 6
6) 3a2 - 7a + 2
7) 3x2 - 13x + 4
8) 4x2 + 4x - 3
9) 5a2 + 2a - 3
10) 5a2 + 13a - 6
11) 6y2 + 5y - 6
12) 6x2 + x - 5
13) 5x2 - 3x - 2
14) 7x2 - 15x + 2
15) 7y2 + 8y + 1
16) 14x2 - 9x + 1
17) 7y2 +18y + 8
18) 9a2 - 3a - 2
19) 8x2 - 26x - 7
20) 3a2 - 5a - 12
21) 3x2 - 10x - 8
22) 6x2 - 5x - 6
23) 4y2 + 25y + 25
24) 7x2 + 20x - 3
25) 5x2 + 2x - 7
26) 10x2 - 11x - 6
27) 15x2 + 14x - 8
28) 8x2 - 26x + 15
29) 12x2 - 7x - 10
30) 9x2 - 12x - 5
31) 8x2 - 2x - 15
32) 10x2 - 21x - 10
33) 15x2 - 26x + 8
40
Difference of Perfect Squares - 6.5
Factor.
1) x2 - 16
2) x2 - 25
3) x2 - 64
4) 9x2 - 1
5) 16x2 - 25
6) 9x2 - 49
7) x4 - 4
8) x8 - 100
9) 36x2 - 1
10) 81x2 - 1
11) 1 - 100x2
12) 1 - 81x2
13) y4 - 121
14) 1 - 144x2
15) x2 + 25
16) x2 + 81
17) x2 - y6
18) x4 - y8
19) 1 - 25x2
20) 1 - 36x2
21) 4 - 9x2
22) 16 - 49x2
23) b2 - 144c2
24) a2 - 49b2
25) x2y2 - 100
26) x6 - 81
27) 9x2 - 16y2
28) 25x2 - 144
29) x2y2 - 1
30) x2 - 400
31) 36a2 - 1
32) 49x2- 4
33) x4- 4
41
Perfect Square Trinomials - 6.5
Factor:
1) x2 + 4x + 4
2) x2 + 8x + 16
3) x2 - 2x + 4
4) x2 - 10x + 25
5) x2 + 16x + 64
6) 9x2 - 6x + 1
7) 36x2 - 12x + 1
8) 100x2 - 20x + 1
9) 4x2 - 40xy + 25y2
10) 25x2 - 30x + 9
11) 49x2 - 14x + 1
12) 49x2 + 70x + 25
13) 16x2 + 40x + 25
14) x2 + 20xy + 100y2
15) x2 - 10xy + 100y2
16) 16x2 - 24xy + 9y2
17) 4x2 - 20xy + 25y2
18) 4x2 + 40xy - 25y2
19) 4x2 + 12xy + 9y2
20) 9x2 - 30xy + 25y2
42
Factor completely.
Factor Completely
1) 3x2 - 12x - 96
2) 3x3 - 18x2 + 15x
3) 5x2 – 80
4) 5x2 - 180
5) 4x2 + 56x + 144
6) 3x2 - 18x + 27
7) 2x2 - 24x + 64
8) 2x2 - 22x + 60
9) 7x2 – 7
10) 3x2 + 6x - 105
11) 4x2 - 100
12) 3x2 + 27
13) 6x3 - 6x
14) x3 - 14x2 + 48x
15) x4 - 6x3 - 7x2
16) x3 - 36x
17) 4x2 - 8x + 28
18) x5 + 14x4 - 32x3
43
Factoring Completely
Factor Completely.
1) 3x2 - 12
2) 2x2 - 50
3) x3 + 2x2+ x
4) y3 - 8y2 +16y
5) x4 + x3 - 6x2
6) a4 - 3a3 - 40a2
7) 3b2 + 30b + 63
8) 5a2+ 7a - 6
9) 4y2 - 32y + 28
10) 2a2 - 18a - 44
11) x3 - 8x2 - 20x
12) b3 - 5b2 - 6b
13) 3x(x - 2) - 5(x - 2)
14) 5a3 - 30a2 + 45a
15) 4x2 - 6x + 2
16) 2x4 - 11x3 + 5x2
17) x4 - 16x2
18) a4 - 81
19) 15x3 - 18x2 + 3x
20) 3ax + 3bx - 3a - 3b
21) 3xy2 + 11xy - 20x
22) 24 + 6x - 3x2
23) a2b2 + 7ab2 - 8b2
24) 4x2y + 12xy + 8y
25) 72 + 2a2
26) 18a3 - 54a2 + 36a
27) 2x2 - 2xy + 4x - 4y
28) 5x2 - 45y2
29) x4 - 9x2
30) 2x2 - 3x + 2xy - 3y
44
Solve.
1) (y+1)(y+2) = 0
Solving Quadratic Equations by Factoring - 6.7.1
2) (y - 4)(y - 6) = 0
3) (z - 6)(z - 1) = 0
4) (x + 7)(x - 5) = 0
5) x(x - 8) = 0
6) x(x + 1) = 0
7) a(a - 4) = 0
8) a(a + 7) = 0
9) y(3y + 2) = 0
10) t(2t - 5) = 0
11) 3a(2a - 1) = 0
12) 2b(4b + 3) = 0
13) (b - 1)(b - 4) = 0
14) (b - 7)(b + 4) = 0
15) x2 - 16 = 0
16) x2 - 4x - 21 = 0
17) x2 + 6x - 16 = 0
18) x2 - 5x = 6
19) x2 - 7x = 18
20) x2 - 8x = 9
21) x2- 5x = 14
22) 2a2 - a = 3
23) 4t2 - 13t = -3
24) 5a2 + 13a = 6
25) 2x2 + 5x = -2
26) x(x+10) = 11
27) y(y - 9) = -18
28) x(x+ 5) = 50
29) x(x - 11) = -30
30) (2x + 3)(x - 1) = 25
31) (z + 1)(z - 9) = 39
STORY PROBLEMS - 7.6
45
PROPORTIONS:
1) Doctor Payne prescribes a patient to take
3 tablets of a medication every four hours.
How many tablets would the patient take in 24
hours?
3) Amy is five feet high. At noon one day
she casts a three foot shadow. She is
standing next to a tree that casts a 19.5 foot
shadow at the same time. How tall is the
tree?
DISTANCE, RATE & TIME:
5) An express train travels 440 miles in the
same amount of time that a freight train
travels 280 miles. The rate of the express
train is 20 mph faster than the freight train.
Find the rate of each train.
7) A car travels 315 miles in the same
amount of time that a bus travels 245 miles.
The rate of the car is 10 mph faster than the
bus. Find the rate of the bus.
WORK:
9) Bill took 40 hours to build the barn on his
property. If Sean had built the barn it would
have been done in 24 hours. How long would
it have taken if they had worked together?
11) Ginny can shovel the driveway after a
snow storm in 24 minutes. Ed uses a plow
and can do it in 8 minutes. How long would it
take them if they worked together?
2) Bob has to pay $9.00 in taxes for every
thousand dollars that his house is worth.
How much would he have to pay if his house
is valued at $275,000?
4) In two minutes a printer can print six
pages. How many pages would be printed
after five minutes?
6) A twin engine plane can travel 1600 miles
in the same time that a single engine plane
travels 1200 miles. The rate of the twin
engine plane is 50 mph faster than the single
engine plane. Find the rate of the twin
engine plane.
8) A helicopter flies 720 miles in the same
amount of time that a plane flies 1520 miles.
The rate of the plane was 200 miles faster
than the rate of the helicopter. Find the rate
for each.
10) Josie can put the ingredients for her
family meal together in forty minutes. Her
husband Jon takes sixty minutes to put
together the same ingredients. How long
would it take if they worked together to
prepare the meal?
12) Sergio and Maria are working on a class
project. Sergio can do it in 30 minutes.
Maria can do it on her own in half the time.
How long would it take if they worked
together?
46
Simplifying radicals - 8.1
Perfect Squares
These numbers have a set of “twins” as
factors:
16 = 4  4
(notice the “twins” as factors) = 4
9 =
33 = 3
4 =2
1 =1
144 = 12
a) Try these:
1) 121 _______
2) 25 _______
3) 49 _______
4) 100 _______
5) 36 _______
6)
7) 64 _______
8)
_______
81 _______
NOT so perfect squares: Choose a set of factors, where one is a perfect square. Look for the
largest perfect square that you can find.
18 = 9  2 =
 2 = 3 2
75 = 25  3 =
 3 =5 3
32 =
16  2 =

2 = 4 2
200 =
100  2 =
 2 = 10 2
b) Try these:
1) 72 _______
2) 12 _______
3)
5) 27 _______
6) 8 _______
7)
_______
_______
4)
_______
8) 45 _______
47
Simplifying Radicals - 8.1 and 8.2
Simplify.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
48
SIMPLIFYING RADICALS WITH VARIABLES - 8.2
In a square root the index is 2
2
x
In a cube root the index is 3
3
x
In a fourth root the index is 4
4
x
To simplify radicals with variables look at the radical as a “jail” with the variables trying to “break
out”. The index indicates how many must be in a group to "break out". For instance, if the index is 3
then there must be 3 of the same thing to escape.
3
x3 =
4
x4 =
3
xxx=x
=x
Take note of this one:
3
x6 =
3
x  x  x  x  x  x = x2 (Notice the square means two groups).
But, watch what happens when there is an extra variable……..
x5 (which really means
2
x5 ) =
x  x  x  x  x = x2 x
To figure the answer without drawing all the x’s, simply divide the index into the exponent. The number of
times the answer comes out evenly, is the exponent of the variable on the outside and the remainder is the
exponent under the radical in the answer.
3
x
4
x16
4
3
( = 1 remainder 1) = x x
3
16
(
= 8, no remainder) = x8
2
x7 (
4
7
= 3 remainder 1) = x3
2
x14 (
x
14
= 3, remainder 2) = x3
4
4
x2
Try these:
1.
x5 ____ 2.
x9 ____
3.
3
x7 ____ 4.
x13 ____
5.
4
x20 ____
6.
3
x17 ____
49
Simplifying Radicals with Variables - 8.2
Simplify.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
50
Radicals and (Rational Exponents) - 8.2 +
Index
Exponent
Exponent
=
=
Index
Radicand
Examples: (Assume all variables are > 0.)
a)
=
d)
=
g)
=(
b)
=
e)
=
=
=
c)
=
f)
=(
) =
) =
Use rational exponents to simplify the following. Assume that variables
represent positive numbers.
1)
3)
2)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
51
Imaginary Numbers
Examples:

i
•

3 = 3i

i




•

6 = 6i


Now Try These:
1)
_______
2)
_______
3)
4)
_______
5)
_______
6)
7)
10)
_______
8)
_______
9)
_______
_______
_______
_______
52
The Pythagorean Theorem – 8.6.1
For each right triangle, find the value of x.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
53
Solving by Taking Square Roots
Solve by taking square roots.
1) x2 = 4
2) y2 = 16
3) v2 - 81 =0
4) Z2 - 144 = 0
5) 9x2 - 25 = 0
6) 16w2 - 81 = 0
7) 16w2 = 25
8) 4x2 = 81
9) 25v2 - 16 = 0
10) 36x2 - 49 = 0
11) 4x2 - 9 = 0
12) 9x2 - 100 = 0
13) x2 + 9 = 0
14) y2 + 100 = 0
15) w2 - 8 = 0
16) v2 - 18 = 0
17) x2 - 50 = 0
18) (x + 1) 2 = 9
19) (y - 2)2 = 36
20) 3(x + 5)2 = 27
21) 5(z - 2)2 = 80
22) 4(x - 1)2 - 25 = 0
23) 9(y + 2)2 - 100 = 0
24) 16(w + 3)2 - 49 = 0
25) 25(y - 1)2 - 36 = 0
26) (x - 3)2 - 32 = 0
27) (y + 4)2 - 75 = 0
28) (x - 1)2 - 50 = 0
29) (x + 1)2 - 80 = 0
Solving Quadratic Equations
by the Quadratic Formula
Determine the value of a, b, and c in the quadratic equation.
54
1. x2 - x - 42= 0
2. x2 + 8x - 20= 0
3. x2 - 10x -24 = 0
4. 2x2 + 3x + 6 = 0
5.
Fill in the a, b, c values in the quadratic equation, but do NOT solve.
6. x2 - x - 42= 0
7. x2 + 8x - 20= 0
8. x2 - 10x -24 = 0
9. 2x2 + 3x + 6 = 0
10.
Solve using the quadratic formula.
11. x2 - x - 42= 0
12. x2 + 8x - 20= 0
13. x2 - 10x -24 = 0
14. 2x2 + 3x + 6 = 0
15.
55
Answers to Worksheet Problems
Worksheet 1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
<
>
>
<
<
<
>
>
<
>
>
<
>
>
>
>
<
<
>
>
<
<
>
>
<
<
>
<
>
>
>
<
>
>
>
<
Worksheet 2
1. -7
2. -11
3. 4
4. 5
5. 18
6. -34
7. 28
8. 77
9. -66
10. 3
11. 3
12. 7
13. 5
14. 4
15. 4
16. 17
17. -4
18. 15
19. 17
20. 16
21. 24
22. -19
23. -21
24. -19
25. -13
26. 26
27. -22
28. -31
29. -35
30. -33
31. 30
32. 21
33. 39
34. 28
35. -33
36. -43
Worksheet 3
A:
1. 13
2. -15
3. 17
4. -20
5. -11
6. 14
7. 23
8. -10
9. 15 10. -12
11. 15 12. -7
13. 12 14. -60
15. 24
B:
1. 1
3. 10
5. 0
7. -3
9. 33
11. 15
13. -5
15. -2
C:
1. 11
3. 0
5. 0
7. -19
9. -17
11. -14
13. 17
15. 26
17. 5
19. 26
2. 5
4. 6
6. 0
8. -30
10. -30
12. 12
14. 0
2. 0
4. -10
6. 13
8. 24
10. 21
12. 20
14.-12
16. 6
18.-18
20. 0
Worksheet 4
1. 12
2. -4
3. -14
4. 7
5. -28
6. -12
7. -9
8. 2
9. -8
10. 14
11. -3
12. 4
13. -50
14. 3
15. 6
16. -12
17. -14 18. 72
19. -16
20. -2
21. 1
22. -10
23. 1
24. 0
25. 24
26. -15
27. -26
28. -25
29. 19
30. -7
31. 44
32. -2
33. 20
34. 25
35. 3
36. 20
37. -32
38. 38
39. -23
40. 9
41. 1
42. 3
43. -4
44. -28
45. 9
46. 0
47. -2
48. -13
49. -6
50. -18
51. -4
52. -19
53. -26 54. -1
55. -9
56. 0
57. -10 58. 1
59. 0
60. 0
Worksheet 5
A.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
6
2
6
2
-3
-3
-9
-9
18
-2
-2
18
B.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
1
-7
15
-1
0
5
-8
8
20
17
-17
7
Worksheet 6
1. -3
2. 20
3. -27
4. 44
5. -13
6. -18
7. -6
8. -4
9. -39
10. -13
11. -17
12. -52
13. 9
14. -9
15. 27
16. -9
17. -20
18. 0
19. -4
20. 36
21. 18
22. -7
23. -61
24. 0
25. 17
26. -55
Worksheet 7
1. -15
2. 24
3. 1
4. 63
5. -50
6. 12
7. 56
8. 27
9. -32
10. 6
11. 42
12. -64
13. 9
14. 35
15. -15 16. 16
17. 108 18. -56
19. 75
20. -96
21. 1
22. -8
23. -36 24. -36
25. 20
26. 2
27. -42 28. -30
29. 24
30. -256
31. -84 32. 30
33. 40
34. -42
35. 125 36. -4
37. 3
38. -24
39. 3
40. -5
41. -42 42. -5
43. -4
44. 72
45. -12 46. 30
47. 100 48. 20
Worksheet 8
1. 4
3. -5
5. 5
7. 4
9. 12
11. -4
13. -6
15. 7
17. 1
19. 4
21. -6
23. -19
25. 4
27. 4
29. -5
31. -20
33. -12
35. -5
37. 9
39. 8
41. 7
43. 26
45. 11
2. -2
4. -9
6. 4
8. -5
10. -3
12. -9
14. -1
16. -7
18. -11
20. -3
22. 23
24. 15
26. -9
28. 5
30. -2
32. -3
34. 3
36. -3
38. -6
40. 3
42. -1
44. 17
46. -5
Worksheet 9
A:
1. 4
2. 9
3. 16
4. 25
5. 36
6. 49
7. 64
8. 81
9. 100 10. 121
11. 144 12. 1
13. 8
14. 27
15. 64 16. 125
17. 1
18. 0
19. 1
20. 1
B:
1. 4
2. 9
3. 16
4. 25
5. 36
6. -8
7. -27
8. -64
9. -125 10. -216
11. 1 12. -1
13. 1 14. -1
15. 1 16. 36
17. 100 18. -1000
19. 100 20. 1000
C:
1. -4
3. -81
5. -1
2. -36
4. -8
6. -64
D:
1. 2
2. 96
3. 1
4. 12
5. -32
6. 18
7. -24
8. 48
9. 48 10. 45
11. –130 12. 150
13. 43 14. 7
15. -56
Worksheet 10
1. 19
2. 8
3. -31
4. -5
5. 6
6. 3
7. 8
8. 10
9. 6
10. 17
Worksheet 11
1. a = 2
2. y = 5
3. k = -3
4. a = 7
5. y = -10
6. z = -2
7. c = 1
8. x = 0
9. y = -8
10. k = -20
11. y = 6
12. x = 16
13. k = -3
14. m = 7
15. y = 18
16. x = 8
17. a = 0
18. n = 6
19. n = 1
20. y = 2
21. x = 3
22. y = 7
23. a = -2
24. k = 3
25. x = 1/4
26. x = 0
27. m = -6
28. y =1/2
29. x =2
30. k =-11
31. x =5
32. k = -5
33. x =5
34. a = 1
35. x = 4
36. k = 3
37. m = -3
38. z =0
39. a =8
40. y =5
Worksheet 12
1. x= 2
2. x = -4
3. x= 3
4. x= -3
5. x= 5
6. x= -6
7. x = -13
8. x = 1
9. x = -6
10. x= -16
11. x = 2
12. x= -28
13. x = 5
14. x = -3
15. x = 0
16. x = 18
17. x = -2
18. x = -1
19. x = 5
20. x =
21. x = -2
22. x= -10
23. x =
24. x =
Worksheet 13
1. 20
2. 5
3. -6
4. -3
5. -1
6. -1
7. -11
8. 4
9. 8
10. 4
Worksheet 14
1.
2.
3.
4.
5.
6.
-19
-3
1
-30
6
3
7.
8. 7
9. -3
11.
10.
12.
11. 6
13. 7
12.
14.
13.
15. -
14. 15. 2
Worksheet 15
Worksheet 16
1.
23.
2. 1
3. -3
24.
4.
25.
5. -2
6. -1
26. 140
27. 4
7.
28.
8. -9
29. 45
9.
30.
10. 5
11. -1
31. -12
32. 0
12.
33.
13. -4
14. -10
15. -2
34. 0
35. -33
36. 0
16.
37.
17. 2
18.
19.
20. -20
21.
22.
38.
39. -6
40.
41.
42.
Worksheet 17
1. x = -y + 12
-3
2. y =
x–6
2
3. b = a – 5
-3
9
4. a =
b+
2
2
5. x = y + 1
6. x = 2y + 10
7. x = -2y + 12
27
8. a = 9b +
2
-b
9. a =
+2
4
1
5
10. y = x 2
4
11. y = -x + 12
-2
12. x =
y+4
3
13. a = b + 5
-2
14. b =
a+3
3
15. y = x - 1
1
16. y = x – 5
2
17. y =
+6
1
3
a9
2
19. b = -4a + 8
5
20. x = 2y +
2
18. b =
Worksheet 21
1. x6
2. b11
3. y10
4. 40x6
5. 25y11
6. -14a7
7. -35y17
8. -24x15
9. -6b6c9
10. -18m2
11. 32x11
12. -28x3y6
13. 48x10
14. x11
15. 14y5z5
16. -15x8y4
17. -64x17
18. 44x13
19. 60c9
20. 4b12
21. a11b15
22. x7y7
23. -7a11
24. 14x13
25. 4a7b4c6
26. 56m3n3
27. -60a7b3
28. -48x6y2
29. 6x7y5
30. -3y10
Worksheet 22
1. 9r4s2
2. 125x9y9
3. -64x9y6
4. 64x6y8
5. 36a6b10
6. 16x16y16
7. 1000x3
8. 81a6
9. 8y15z3
10. -64a3b6
11. a9b9
12. -125x3y6
13. x30y12
14. 4a2b16
15. 9a6
16. x25
17. z21
18. a6
19. m15n20
20. x2y4
21. -p21q21
22. -a25b30
23. v36
24. b9c9
25. a5b5c15
26. x40y16
27. -a20b20
28. -a28c28
29. x12y12z8
30. d30
31. 256x12
32. f13
33. 343x6y9
34. x10y10
35. -125a15b12
36. -m35n14
Worksheet 23
1. 64x9y3
2. 216x14y9
3. 4a7
4. -3x5y10
5. -1152x12
6. -32x5
7. 64a4b10
8. 64x13y2
9. -512x11y6
10. 4m4n8
11. 25b21
12. x8y14
13. x12y12
14. -3x4y5
15. -64a3b11
16. 50x7y15
17. 64x7y5
18. 256a8b10
19. -27x7y31
20. 16x22
21. -2m17n
22. 128p7q7
23. -27x35y54
24. r30s24
Worksheet 24
1. 1
2. 1
3. 1
4. 1
5. 1
6. 1
7. b
8. c
9. x
2
10. a
Worksheet 25
1. x
2. a
3. -c3
4.
5.
6.
2.
4.
3.
5.
6.
4.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
17.
19. 1
20.
18. m9n3
19. x9y7
21.
22.
20.
23.
24.
23. -a2b5
21. a14b2
22. 13a3b5
23. m2n2
25.
26.
24. -c12
24.
27.
28.
25. a9b5
29. y
31. 1
7.
8.
9.
10.
12.
12.
a x
3x
-x
1
3
-3
1.
3.
11.
13.
14.
15.
16.
17.
18.
1.
Worksheet 27
2.
11.
2 2
Worksheet 26
13.
15.
14.
16.
17.
19.
21.
22.
18.
20.
5. 5y3
6. 8x4
7. z3
8. x7
9. m6
10. 4x4
11. 12z6
12. a7b7
13. a10b8
14. x3y10
15. r9s3
16. a9
26.
27. x3y7
28. 8ab4
6
30. a
32. 1
8
Worksheet 28
Worksheet 29
Worksheet 30
Worksheet 31
1. -x2 + 2x
2. y2 + y -5
3. 2x2 + 3x + 3
4. x3 + 3x2 -6x
5. y3 -3
6. x3 -x2 + 4x
7. -9x -5
8. 3x3+2x2+3x+ 1
9. 4y3 -y2 + 4y -7
10. x3 -4x -1
11. 2y3 -12xy -5
12. 3x2 -5xy
13. x2 -5x + 2
14. 4x2 + 7x -8
15. 3x3 -2x2 -8x -5
1. x2 + x
2. 2y - y2
3. -x2 - 2x
4. -8y + y2
5. 2a2 - 2a
6. 3b2 + 15b
7. -2x3 + 2X2
8. -4y3 - 24y2
9. -6y4+ 6y3
10. -2x4 + 3x2
11. 10x3 - 4x2
12. 6y2 - 3y3
13. 8x2 - 12x
14. 2y2 - y
15. 2x2 - 3x
16. 6x2 - 3x
17. -x3y + x2y3
18. -2x2y2 + xy3
19. x3 - 2x2 + x
20. x3 - 3x2 - 2x
21. -y3 + 4y2 - 3y
22. -y3 + 5y2+ 6y
23. -a3 + 6a2 + a
24. -2b3 - 3b2 + 6b
25. 2x4 - 3x3 - 2x2
26. -3y4 - 5y3 - 3y2
27. -x5 - 5x4 - 6x3
28. -2y5 - 3y4 - 4y3
29. -4y4-10y3+16y2
30. 12x4- 6x3+21x2
31. 20x4- 4x3- 36x2
32. -5y4+15y3-30y2
33. x3y - x2y2 + xy3
34. a3b-3a2b2-4ab3
35. x3y-2x2y2+2xy3
36.a3b+5a2b2-7ab3
1. x2 + 5x + 4
2. y2 + 5y + 6
3. a2 + 3a - 10
4. b2 - b - 20
5. y2 - 5y - 14
6. x2 + 5x - 36
7. y2 - 8y + 12
8. a2 - 15a + 56
9. a2 - 10 a + 16
10. x2 + 8x - 33
11. 2x2 + 13x + 6
12. 3y2 + 5y + 2
13. 2x2 + 3x - 9
14. 5x2 + 13x - 6
15. 3x2 -17x + 10
16. 6x2 - 13x + 5
17. 2y2 - 7y - 9
18. 4y2 + y - 14
19. 9x2 + 33x + 28
20. 30a2 + 17a + 2
21. 12a2 - 56a + 65
22. 10a2 - 53a + 63
23. 15b2 + 43b - 44
24. 12a2 + 31a - 30
25. x2 + 3xy + 2y2
26. 2a2+ 5ab + 2b2
27. 2x2 - 5xy + 3y2
28. 2a2 - 3ab - 9b2
29. 8a2+18ab - 5b2
30. 2x2 + xy - y2
31. 9x2 - 9xy - 10y2
32. 30x2+17xy+2y2
1. x2- 1
2. x2 - 9
3. x2 - 25
4. x2 - 49
5. 4x2-1
6. 9x2 - 1
7.16x2 - 9
8. x2 + 10x + 25
9. y2 - 8y + 16
10. 9y2 - 6y + 1
11. x2 - 2x + 1
12. x2 - 6x + 9
13. x2 + 14x + 49
14. x2 + 18x + 81
15. x2 - 2xy + y2
16. 4a2 - 20a + 25
17. 25x2 - 40x + 16
18. 9x2 - 42x + 49
19. 9a2 - 25
20. 36x2 - 25
21. 4x2 + 20x + 25
22. 81x2 - 36x + 4
23. a2 - 4ab + 4b2
24. x2 + 4xy+ 4y2
25. 25x2 - 36
26. b2 - 36a2
27. x2+10xy +25y2
28. 4 - 28y + 49y2
29. 9 - 30y + 25y2
30. 9 - 25y2
31. 16x2 - 1
32. 4a2+12ab+9b2
33. x2+12xy +36y2
Worksheet 32
1. 9x2 - 3x
2. 3x2 - 4x - 4
3. 6x2 - x -1
4. 4x2 + 31x + 21
5. x2 + 6x + 9
6. x2 -16x + 64
7. 4x2 + 4x + 1
8. 9x2 - 24x + 16
9. π(x2 + 8x + 16)
10. π(x2 - 6x + 9)
11. π(x2+ 12x+ 36)
12. π(4x2 + 4x + 1)
Worksheet 33
1. x + 1
2. y + 1
3. 2a - 3
4. 3a - 7
5. 2a - 3
6. 2b - 5
7. 2a + 3
8. 5y + 3
9. 3b2 - 2
10. 2y - 1
11. x2 + 2x - 4
12. a2 - 4a + 5
13. x2 - 2x - 1
14. a3 - 6a2 - 4a
15. xy - 3
16. xy + 5
17. -2y2 + 5
18. -4x2 + 3
19. -y + 4
20. -2 + 3x2
21. - 7x + 4 22. 4x + 1 23. 6y + 4 24. -2x + 5 25.
26.
27.
28.
4a - 5 + 6b
2xy - 1 - 3y
2a - 3 + 5b
2a + 1 - 4b
Worksheet 34
Worksheet 35
1. 4(a + 1)
2. 6(c - 1)
3. 4(2 - a2)
4. 3(3 + 7x2)
5. 3(x + 3)
6. 5(2a2 + 3)
7. 8(3a - 1)
8. 12(2x + 1)
9. 3(3x - 2)
10. 8a(2a - 1)
11. 4y(3x - 4)
12. b2 (6 - 5b)
13. 4x2 (5x - 6)
14. 12a2 (a3 - 3)
15. 6a2b2(4ab2 - 3)
16. 2ab(2a4 + 3b3)
17. a3b2 (1 + ab)
18. 5x2y(5y - 3x)
19. xy(3x - 5y)
20. 4a2b2 (2a - 3b)
21. x2y2 (x - y2)
22. x(x2 - 5x + 7)
23. y(y2 - 6y - 8)
24. 4(x2 - 4x + 5)
25. 3(2y2 - 3y + 4)
26. 3(x2 - 3x + 6)
27. b2(b2 - 3b + 7)
28. 4x2(1 -2x +3x2)
29. 4(3y2- 4y + 12)
30. 5y(y3 + 2y2 - 7)
31. 4x2(x2+ 3x - 7)
32. 15ab(3a3b
- 5a2+2b3)
33.
16x2y2(2x2-6y2 -3x4)
1. (a + b)(x + 3)
2. (x - y)(a + 5)
3. (b - 1)(x - y)
4. (c - d)(a + b)
5. (a - 1)(y - 1)
6. (y + 3)(a -1)
7. (y - 2)(x - 1)
8. (Y - 7)(3x -1)
9. (x - 5)(2x -1)
10. (x - 3)(5y + 1)
11. (a - 2b)(x + y)
12. (a + 1)(4a - 1)
13. (x - 9)(a - 1)
14. (a - 5)(b - 2)
15. (x - 3y)(c + d)
16. (a + 1)(3x + 4)
17. (x -1)(a - 3)
18. (a - 4)(x - y)
19. (a - 3b)(x - y)
20. (a - 9)(x + 1)
21. (x - 7)(a - 3)
22. (x - 6)(2x -1)
23. (e - 5)(d - 1)
24. (a - 4) (2x + y)
25. (a - 9)(m + n)
26. (n - 2)(3m + 1)
27. (b -1)(2a - c)
28. (a - 2)(x - 3y)
29. (y - 5)(x -1)
30. (a - b)(2 - c)
31. (n - 1)(m - 2)
32. (c - 7)(x + y)
33. (b - c)(a + 3)
Worksheet 36
Worksheet 37
Worksheet 40
Worksheet 41
1. (y - 3)(2x + 3)
2. (x + y)(x + 2)
3. (a + 5)(x + 6)
4.(2x + 1)(x - y)
5. (2x+2y)(3x+1)
= 2(x+ y)(3x+1)
6. (4a + 3)(a + b)
7. (a - 2)(x - 7)
8. (2x - 5)(x + y)
9. (a - 2)(x + l)
10. (y - 3)(3x + 1)
11. (a - 2b)(3a - 1)
12. (x - 3)(2a + 1)
13. (x - 2)(2a - 3)
14. (5a - 3x)(a - 3)
15. (x - y)(y - 3)
16. (7a - y)(a - b)
17. (2x - y)(3x - 2)
18. (4a - b)(a + 3)
19.
(a - 4x)(2a + 3)
20. (x - 2)(3x - y)
21. (2a +y)(4x - 1)
1. (a - 7)(a + 5)
2. (a - 3)(a - 1)
3. (a + 5)(a - 2)
4. (a - 3)(a - 2)
5. (b - 5)(b - 2)
6. (b + 5)(b + 3)
7. (y + 11)(y - 6)
8. (x - 10)(x + 6)
9. (y - 2)(y - 5)
10. (y - 6)(y - 3)
11. (x - 6)(x - 6)
12. (x - 12)(x + 8)
13. (a + 7)(a - 4)
14. (x + 8)(x + 2)
15. (b - 20)(b + 9)
16. (x + 5)(x + 5)
17. (x - 7) (x - 7)
18. (b + 3)(b + 4)
19. (b + 8)(b + 2)
20. (x - 12)(x + 3)
21. (x - 12)(x + 5)
22. (x + 14)(x - 4)
23. (x - 16)(x + 8)
24. (x - 11)(x + 7)
25. (b - 14)(b - 6)
26. (b - 9)(b - 12)
27. (b - 12)(b - 15)
28. (a + 7)(a + 9)
29. (x - 15)(x - 4)
30. (x - 21)(x - 4)
1. (x - 2)(2x - 1)
2. (3x + 1)(x - 1)
3. (2a + 1)(a + 3)
4. (3x - 2)(x + 1)
5. (2b - 1)(b - 6)
6. (3a - 1)(a - 2)
7. (3x - 1)(x - 4)
8. (2x + 3)(2x - 1)
9. (5a - 3)(a + 1)
10. (5a - 2)(a + 3)
11. (3y - 2)(2y + 3)
12. (6x - 5)(x + 1)
13. (5x + 2)(x - 1)
14. (7x - 1)(x - 2)
15. (7y + 1)(y + 1)
16. (7x - 1)(2x - 1)
17. (7y + 4)(y + 2)
18. (3a + 1)(3a - 2)
19. (4x + 1)(2x - 7)
20. (3a + 4)(a - 3)
21. (3x + 2)(x - 4)
22. (3x + 2)(2x - 3)
23. (4y + 5)(y + 5)
24. (7x - 1)(x + 3)
25. (5x + 7)(x - 1)
26. (5x + 2)(2x - 3)
27. (5x - 2)(3x + 4)
28. (4x - 3)(2x - 5)
29. (4x - 5)(3x + 2)
30. (3x - 5)(3x + 1)
31. (4x + 5)(2x - 3)
32. (5x + 2)(2x - 5)
33. (5x - 2)(3x - 4)
1. (x + 4)(x - 4)
2. (x + 5)(x - 5)
3. (x + 8)(x - 8)
4. (3x + 1)(3x - 1)
5. (4x + 5)(4x - 5)
6. (3x + 7)(3x - 7)
7. (x2 + 2)(x2 - 2)
8. (x4+10)(x4-10)
9. (6x + 1)(6x - 1)
10. (9x + 1)(9x -1)
11. (1+10x)(1-10x)
12. (1 +9x)(1 -9x)
13. (y2+11)(y2-11)
14. (1+12)(1-12x)
15. Irreducible over
the integers
16. Irreducible over
the integers
17. (x + y3)(x - y3)
18. (x2+ y4)
(x- y2)(x+ y2)
19. (1 + 5x)(1 - 5x)
20. (1 +6x)(1 -6x)
21. (2 + 3x)(2 - 3x)
22. (4 + 7x)(4 - 7x)
23. (b+12c)(b-12c)
24. (a +7b)(a -7b)
25. (xy+10)(xy -10)
26. (x3 + 9)(x3 - 9)
27. (3x+4y)(3x-4y)
28. (5x+12)(5x-12)
29. (xy + 1)(xy - 1)
30. (x + 20)(x - 20)
31. (6a + 1)(6a - 1)
32. (7x + 2)(7x - 2)
33. (x2 + 2)(x2 - 2)
Worksheet 42
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
(x + 2)2
(x + 4)2
Prime
(x - 5)2
(x + 8)2
(3x - 1)2
(6x - 1)2
(10x - 1)2
(2x – 5y)2
(5x - 3)2
(7x - 1)2
(7x + 5)2
(4x + 5)2
(x + 10y)2
Prime
(4x + 3y)2
(2x - 5y)2
Prime
(2x + 3y)2
(3x - 5y)2
Worksheet43
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
3(x – 8)(x + 4)
3x(x – 5)(x – 1)
5(x + 4)(x – 4)
5(x + 6)(x – 6)
4(x2 + 14x + 36)
3(x – 3)(x – 3)
2(x – 8)(x – 4)
2(x – 6)(x – 5)
7(x + 1)(x – 1)
3(x + 7)(x – 5)
4(x + 5)(x – 5)
3(x2 + 9)
6x(x + 1)(x – 1)
x(x – 8)(x – 6)
x2 (x – 7)(x + 1)
x(x + 6)(x – 6)
4(x2 – 2x + 7)
x3(x + 16)(x – 2)
Worksheet 44
1. 3(x+2)(x-2)
2. 2(x-5)(x+5)
3. x(x+1) 2
4. y(y -4)2
5. x2(x+3)(x-2)
6. a3(a-8)(a+5)
7. 3(b+3)(b+7)
8. (5a-3)(a+2)
9. 4(y-1)(y-7)
10. 2(a-11)(a+2)
11. x(x-10)(x+2)
12. b(b-6)(b+1)
13. (3x–5)(x–2)
14. 5a(a-3)2
15. 2(2x-1)(x-1)
16. x2(2x-1)(x-5)
17. x2(x+4)(x-4)
18. (a2+9)
(a+3)(a-3)
19. 3x(5x-1)(x-1)
20. 3(x-1)(a +b)
21. x(3y-4)(y+5)
22. -3(x-4)(x+2)
23. b2(a+8)(a-1)
24. 4y(x+2)(x+1)
25. 2(36+a2)
26. 18a(a-2)(a-1)
27. 2(x+2)(x-y)
28. 5(x+3y)(x-3y)
29. x2(x+3)(x-3)
30. (x+y)(2x-3)
Worksheet 45
1. -1, -2
2. 4, 6
3. 6, 1
4. -7, 5
5. 0, 8
6. 0, -1
7. 0, 4
8. 0, -7
9. 0, 10. 0,
11. 0,
12. 0,
13.
14.
15.
16.
17.
18.
19.
20.
21.
1, 4
-4, 7
-4, 4
-3, 7
-8, 2
-1, 6
9, -2
-1, 9
-2, 7
22. -1,
23.
,3
24. -3,
25. 26.
27.
28.
29.
, -2
-11, 1
3, 6
-10, 5
6, 5
30. -4,
31. -4, 12
Worksheet 46
18 tablets
$2475
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15 pages
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9. 15 hrs
10. 24 min
11. 6 min
12. 10 min
1.
2.
3.
4.
5.
Worksheet47
a)
1. 11
2. 5
3. 7
4. 10
5. 6
6. 3
7. 8
8. 9
b)
1.
2.
3.
4.
5.
6.
7.
8.
Worksheet 48
1. 5
2. 11
3. 13
4. 14
5. 17
6. 20
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18. 36
19.
20. 30
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Worksheet 49
1.
2.
3.
4.
5.
6.
Worksheet 50
1. x
2. x
4
2.
4.
3
3
5
2
5. x y
6. x y
7. 3x
3.
5.
6. i
7.
4.
3
8. 4 y
9. 6x
8.
5.
5
9.
10.
6.
10.
11.
7.
12.
8.
13.
14.
9.
15.
10.
16.
17. 6ab
3
11.
2 8
=
18. 10x y
19.
20.
12.
=
13.
=
21.
14.
22.
=
2 2
23. x y
=
24.
25.
15.
26.
27.
28. 10(a + 5)
29.
30.
31.
32.
33.
Worksheet 52
1. 9i
2. 7i
3.
1. 3
5
3.
4.
Worksheet 51
8(x + y)
7(x - 3)
2(x - 1)
x+6
b-5
2
2
=
Worksheet 53
1. 5
2. 15
3. 5
4.
5.
6.
7.
8. 100
9.
10.
Worksheet 54
Worksheet 55
5.
,
6.
,
7.
,
8.
,
1. a
c
2. a
c
3. a
c
4. a
c
5. a
9.
,
6.
10.
,
11.
,
1. -2, 2 2. -4, 4
3. -9, 9 4. -12.12
12.
-4, 2
-4, 8
-2, -8
6, -2
,
23.
,
24.
,
25.
,
27.
28.
29.
b = +8;
b = -10;
b = +3;
b = -11; c = +10
8.
9.
16.
26.
b = -1;
,
15.
22.
+1;
-42
+1;
-20
+1;
-24
+2;
+6
+2;
7.
13. No real number
solution
14. No real number
solution
17.
18.
19.
20.
21.
=
=
=
=
=
=
=
=
=
10.
11. x = +7; x = -6
12. x = -10; x = +2
13. y = -2; y = +12
14.
15.