Review Problems for Basic Algebra II Students. Note: It is very ... Translate into an
algebraic expression: three more than half of a .... worksheet. For example:.
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))
Examples
What
you
will
need
Nothing!
The
answer
is…
Easy!
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
What
you
will
need
The
answer
is…
Given:
m
(slope)
and
a
point
The
y‐intercept
or
b.
(that
is
not
the
y
intercept)
Use
the
m
given
• Use
the
m
that
is
given
• Use
the
point
(x,
y)
Use
the
b
you
found
• Replace
the
x,
m,
and
y
Discard
the
point
(x,
y)
• y
=
mx
+
b
(
)
=
(
)(
)
+
b
Write
the
equation
of
the
line!
• Solve
for
b
Examples
Given:
m
=
5,
(2,
7)
y
=
m
x
+
b
(
)
=
(
)(
)
+
b
Y
=
5x
‐
3
7
=
(5)(2)
+
b
7
=
10
+
b
‐3
=
b
Y
=
mx
+
b
2 2 Given:
m
=
,
(3,
‐
2)
Y
=
x
‐
4
(
)
=
(
)(
)
+
b
3 3 2 ‐2
=(
)(3)
+
b
3 ‐2
=
2
+
b
‐4
=
b
Find
the
Equation
of
the
line
given
the
following
information:
2 1 5
6.
m
=
‐2,
(3,
3)
7.
m
=
5,
(
,
5)
8.
m
=
,
(‐5,
0)
9.
m
=
‐7,
(2,
4)
10.
m
=
,
(7,
2)
5 5 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)
Given:
(3,
2)
and
(‐3,
6)
Using
the
Point‐Slope
Formula:
y
–
y1
=
m
(x
–
x1)
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
=
m
x
+
b
(
)
=
(
)(
)
+
b
5
=
(‐1)(‐2)
+
b
5
=
2
+
b
3
=
b
m=
y2 - y1 6 - 2 4 -2 = = = x2 - x1 -3 - 3 -6 3
y
–
y1
=
m(x
–
x1)
‐2
y
–
(2)
=
(x
–
3)
3 ‐2
y
–
2
=
x
+
2
3 ‐2
y
=
x
+
4
3
Discard
both
points
Write
the
equation
of
the
line!
Y
=
‐
x
+
3
‐2
y
=
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 1.
y
=
‐2x
+
5
6.
y
=
‐2x
+
9
2.
y
=
5x
+
9
7.
y
=
5x
+
3
1 3.
y
=
+
5
5 1 8.
y
=
x
+
1
5
4.
y
=
‐7x
‐
2
9.
y
=
‐7x
+
18
11.
y
=
‐2x
+
5
12.
y
=
‐2x
–
7
13.
y
=
3x
+
8
14.
y
=
2x
–
2
5 5.
y
=
x
7
5 10.
y
=
x
‐
3
7
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 =
33 = 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
xxx=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.
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