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. Review Problem Reference section in text Answer 1.

Simplify: 1618

0.1 89

2. Combine:

38 +

56

0.2 29 24

3. Combine:

57 -

29

0.2 3163

4. Multiply:

712 x

828

0.3 16

5. Divide:

614 ÷

38

0.3 87

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. Review Problem Reference section in text Answer 1. Combine: 7 + (- 6) – 3

1.2 - 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 3 +

12 x

10. Explain how you would locate the point (4, -3) on graph paper.

3.1 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 0

15) 7 -6

16) 9 -7

17) -12 14

18) -10 3

19) 30 27

20) 33 16

21) -24 42

22) -34 47

23) 19 -31

24) 43 -36

25) -37 -29

26) -41 -27

27) 53 -71

28) -90 70

29) 53 -64

30) 91 -67

31) -53 -81

32) -88 -67

33) 84 73

34) 67 59

35) 48 -37

36) -55 53 1

The Opposite of a Number

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) =

(B) 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 =

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 Multiply: 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) =

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) 2) 3) 4) 5)

6) 7) 8) 9) 10)

11) 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)

8 EXPONENTS - 1.4

A. Raise each base to its given power:

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 = ____ 10) 5(-3)2 = ____ 11) -92 -72 = ____ 12) 53 + 52 = ____ 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

8) 32 ÷ 42

7 - 5 - (-6)

9) 11 + 19 - 332 - 1 - 7

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 = 5y - 12 + 3y

2y - 6 = 8y - 12 - 2y +12 -2y +12 6 = 6y 1 = 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

13 Equations With Parentheses - 2.3

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:

1) x + = 5 2) 3x + 5 = x + 7

3) (x + 6) = x + 4 4) x = - x

5) 4x - 3 = x - 9 6) 3 - x = 2(1 - x)

7) x + = 8) 3x - 1 = 2(x - 5)

9) 2x - = x 10) 3x + 2 = 5x - 8

11) 3(x + 4) + 1 = 9 - x 12) 2x - = x + 2

13) x - 7 = 4x + 5 14) (2x + 4) = (x - 5)

15) 2x +3 = 3x + 5 16) 4x - = x + 3

17) x - 6 = 5x -14 18) x + = + x

19) x - 5 = x + 1 20) 3x + 7 = 2x + (2x +1)

21) 5x + = 5 - x 22) 3x + 7 = 5x - 4

15 Supplementary Equations (Cont.)

Solve:

23) 24) 3x + 2(x - 5) = 7 - (x + 3)

25) 3 - x = (7 + 2x) 26) x - = x + 3

27) 5x - 3(x + 1) = 5 28) x + = 3x +

29) x + 6 - x = x + 9 30) 7x + 5 = 2(x - 1) - 21

31) (x + 3) + 8 = x + 11 32) 5 + 6x - 3 = 2 + 4x

33) x = 11 + x - 34) x + = - (2x - 5)

35) 5x - (2x - 3) = 4(x + 9) 36) (x + 6) = (10 - x)

37) 6x - = x + 4 + x 38) =

39) (4 - x) - 6 = x + 6 40) x = 28 - x

41) (x + 6) + 5 = 2x + 28 42) 2x + = x + 1

16

Literal Equations - 3.1

Literal Equations ~ Equations that contain more than one letter Example: 2x + 3y = 12. Solve for x.

2x = - 3y + 12 2x2 =

-3y2 +

122

x = -32 y + 6

Example: 2x + 3y = 12. Solve for y.

3y = - 2x + 12 3y3 =

-2x3 +

123

y = -23 x + 4

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) x2 + y = 6. Solve for x. 8)

23 a - 6b = 9. Solve for a.

9) a + b4 = 2. Solve for a.

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) x2 + y = 6. Solve for y. 18)

23 a - 6b = 9. Solve for b.

19) a + b4 = 2. Solve for b.

20) 2x - 4y = 5. Solve for x. 17

Equation: y = 3x - 2

x y

0

1

2

Equation: y = -2x + 4

x y

0

1

2

T a b l e o f V a l u e s

Equation:

x y

0

3

6

18

Equation: y = x2 - 4

x y

-3

-2

-1

0

1

2

3

Equation: y = x2 - x - 2

x y

-3

-2

-1

0

1

2

3

T a b l e o f V a l u e s

Equation: y = x2 +2x - 6

x y

-3

-2

-1

0

1

2

3

19

Equation:

x y

Equation:

x y

T a b l e o f V a l u e s

Equation:

x y

20

The Equation of a Line y = mx + b

FindtheEquationofthelinegiventhefollowinginformation:

AInformationgiven Whatyouwillneed Theansweris…Given:m(slope)andb(y‐interceptor(0,b))

Nothing!

Easy!

ExamplesGiven: m = 3, b = 7

Y = 3x + 7

Given: m = 23 , (0, - 4)

Y =

23 x - 4

FindtheEquationofthelinegiventhefollowinginformation:

1. m=‐2,b=252.m=5,(0,9)3.m=

15,b=54.m=‐7,b=‐25.m=

57,(0,0)

***********************************************************************************

BInformationgiven Whatyouwillneed Theansweris…Given:m(slope)andapoint(thatisnottheyintercept)

They‐interceptorb.• Usethemthatisgiven• Usethepoint(x,y)• Replacethex,m,andy• y=mx+b

()=()()+b• Solveforb

UsethemgivenUsethebyoufoundDiscardthepoint(x,y)Writetheequationoftheline!

ExamplesGiven:m=5,(2,7)

y=mx+b()=()()+b7=(5)(2)+b7=10+b‐3=b

Y=5x‐3

Given:m=23,(3,‐2)

Y=mx+b()=()()+b

‐2=(23)(3)+b

‐2=2+b‐4=b

Y=23x‐4

FindtheEquationofthelinegiventhefollowinginformation:

6.m=‐2,(3,3)7.m=5,(25,5)8.m=

15,(‐5,0)9.m=‐7,(2,4)10.m=

57,(7,2)

20A

CInformationgiven Whatyouwillneed: Theansweris…Given:Twopoints(x1,y1)(x2,y2)

1. Theslope:m=y2‐y1x2‐x1

2. They‐interceptorb.

UsethemyoufoundUsetheb youfound

Now…• Findm• Useoneofthepoints(x,y)• Replacethex,m,andy• y=mx+b

()=()()+b• Solveforb

DiscardbothpointsWritetheequationoftheline!

ORUsethe

Point‐SlopeFormula:y–y1=m(x–x1)

ExamplesGiven:(‐2,5)and(4,‐1)

1.Findtheslope:

m=y2‐y1x2‐x1

=‐1‐54‐(‐2)=

‐66 =‐1

2.Now,useonlyoneofthepoints.y=mx+b()=()()+b5=(‐1)(‐2)+b5=2+b3=b

Y=‐x+3

Given:(3,2)and(‐3,6)UsingthePoint‐SlopeFormula:

y–y1=m(x–x1)

m = y2 - y1 x2 - x1

= 6 - 2-3 - 3 =

4-6 =

-23

y–y1=m(x–x1)

y–(2)=‐23 (x–3)

y–2=‐23 x+2

y=‐23 x+4

y=‐23 x+4

FindtheEquationofthelinegiventhefollowinginformation: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:

1.y=‐2x+25

2.y=5x+9 3.y=

15+5

4.y=‐7x‐2 5.y=

57x

6.y=‐2x+9

7.y=5x+3 8.y=

15x+1

9.y=‐7x+18

10.y=57x‐3

11.y=‐2x+5

12.y=‐2x–7

13.y=3x+8

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. (2a2)4 = (2a2) (2a2) (2a2) (2a2) = 16a8

b. If there is no numerical coefficient, multiply the exponents inside the parentheses by the exponent that is outside the parentheses. (a3b5)6 =x18y30

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

22 Multiplication of Monomials and Monomials (Exponents outside the Parentheses) - 5.1

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 c. d. a3boc = a3c

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) 20)

21) 22) 23) 24)

25) 26)

27) 28)

26

Negative Exponents (Cont.) - 5.2.1 Write with a positive exponent. Then evaluate. 1) 2) 3) 4) 5) 6) 7) 8)

9) 10) 11) 12)

Simplify. 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24)

25) 26) 27) 28)

29) 30) 31) 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)

(4x2 + 2x2) + (3x - 5x) + (-2 -6) 6x2 - 2x - 8

b. (3a2 -5a + 2) - (4a2+ a + 2) 3a2 - 5a + 2 - 4a = -a -2 (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. 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. 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. 7. The length of a side of a square is 2x + 1. Find the area of the square m 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. 11. The radius of a circle is x + 6. Find the area of the circle 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. 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. 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. 8. The length of a side of a square is 3x - 4. Find the area of the square 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. 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 34

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

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

Solving Quadratic Equations by Factoring - 6.7.1 Solve. 1) (y+1)(y+2) = 0 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

45

STORY PROBLEMS - 7.6

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?

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?

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?

4) In two minutes a printer can print six pages. How many pages would be printed after five minutes?

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.

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.

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.

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.

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?

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?

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?

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

32 = 16 2 = 2 = 4 2

75 = 25 3 = 3 = 5 3

200 = 100 2 = 2 = 10 2

b) Try these: 1) 72 _______ 2) 12 _______ 3) _______ 4) _______ 5) 27 _______ 6) 8 _______ 7) _______ 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 = 3

x x x = x

4

x4 = = 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.

3x4 (

43 = 1 remainder 1) = x

3x x7 (

72 = 3 remainder 1) = x3 x

x16 (162 = 8, no remainder) = x8 4

x14 ( 144 = 3, remainder 2) = x3

4x2

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) =

b) = c) =

d) =

e) = = = f) = ( ) =

g) = ( ) =

Use rational exponents to simplify the following. Assume that variables represent positive numbers. 1)

2) 3) 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) _______ 8) _______ 9) _______

10) _______

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

54

Solving Quadratic Equations by the Quadratic Formula

Determine the value of a, b, and c in the quadratic equation.

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 2. 5 3. 10 4. 6 5. 0 6. 0 7. -3 8. -30 9. 33 10. -30 11. 15 12. 12 13. -5 14. 0 15. -2 C: 1. 11 2. 0 3. 0 4. -10 5. 0 6. 13 7. -19 8. 24 9. -17 10. 21 11. -14 12. 20 13. 17 14.-12 15. 26 16. 6 17. 5 18.-18 19. 26 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. 6 2. 2 3. 6 4. 2 5. -3 6. -3 7. -9 8. -9 9. 18 10. -2 11. -2 12. 18 B. 1. 1 2. -7 3. 15 4. -1 5. 0 6. 5 7. -8 8. 8 9. 20 10. 17 11. -17 12. 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 2. -2 3. -5 4. -9 5. 5 6. 4 7. 4 8. -5 9. 12 10. -3 11. -4 12. -9 13. -6 14. -1 15. 7 16. -7 17. 1 18. -11 19. 4 20. -3 21. -6 22. 23 23. -19 24. 15 25. 4 26. -9 27. 4 28. 5 29. -5 30. -2 31. -20 32. -3 33. -12 34. 3 35. -5 36. -3 37. 9 38. -6 39. 8 40. 3 41. 7 42. -1 43. 26 44. 17 45. 11 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 2. -36 3. -81 4. -8 5. -1 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

11.

12.

13. 7

14.

15. -

Worksheet 14 1. -19 2. -3 3. 1 4. -30 5. 6 6. 3

7.

8. 7 9. -3

10.

11. 6

12.

13.

14. -

15. 2

Worksheet 15

1.

2. 1 3. -3

4.

5. -2 6. -1

7.

8. -9

9.

10. 5 11. -1

12.

13. -4 14. -10 15. -2

16.

17. 2

18.

19.

20. -20

21.

22.

Worksheet 16

23.

24.

25.

26. 140 27. 4

28.

29. 45

30.

31. -12 32. 0

33.

34. 0 35. -33 36. 0

37.

38.

39. -6

40.

41.

42.

Worksheet 17 1. x = -y + 12

2. y = -32 x – 6

3. b = a – 5

4. a = -32 b +

92

5. x = y + 1 6. x = 2y + 10 7. x = -2y + 12

8. a = 9b + 272

9. a = -b4 + 2

10. y = 12 x -

54

11. y = -x + 12

12. x = -23 y + 4

13. a = b + 5

14. b = -23 a + 3

15. y = x - 1

16. y = 12 x – 5

17. y = + 6

18. b = 19 a -

32

19. b = -4a + 8

20. x = 2y + 52

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 10. a2

11.

12.

13. a2x2 14. 3x 15. -x 16. 1 17. 3 18. -3

Worksheet 25 1. x 2. a

3. -c3 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21.

22.

23. -a2b5

24. -c12

Worksheet 26

1.

2.

3.

4.

5. 5y3 6. 8x4 7. z3 8. x7 9. m6 10. 4x4 11. 12z6 12. a7b7 13. a10b8 14. x3y10 15. r9s3 16. a9

17.

18. m9n3 19. x9y7

20.

21. a14b2 22. 13a3b5 23. m2n2

24.

25. a9b5

26.

27. x3y7 28. 8ab4

Worksheet 27

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 1 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. y6 30. a8 31. 1 32. 1

Worksheet 28

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

Worksheet 29 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

Worksheet 30 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

Worksheet 31

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. 4a - 5 + 6b 26. 2xy - 1 - 3y 27. 2a - 3 + 5b 28. 2a + 1 - 4b

Worksheet 34 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)

Worksheet 35

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

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)

Worksheet 37 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)

Worksheet 40 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)

Worksheet 41

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. (x + 2)2 2. (x + 4)2

3. Prime 4. (x - 5)2

5. (x + 8)2

6. (3x - 1)2

7. (6x - 1)2

8. (10x - 1)2

9. (2x – 5y)2

10. (5x - 3)2

11. (7x - 1)2

12. (7x + 5)2

13. (4x + 5)2

14. (x + 10y)2

15. Prime 16. (4x + 3y)2

17. (2x - 5y)2

18. Prime 19. (2x + 3y)2

20. (3x - 5y)2

Worksheet43 1. 3(x – 8)(x + 4) 2. 3x(x – 5)(x – 1) 3. 5(x + 4)(x – 4) 4. 5(x + 6)(x – 6) 5. 4(x2 + 14x + 36) 6. 3(x – 3)(x – 3) 7. 2(x – 8)(x – 4) 8. 2(x – 6)(x – 5) 9. 7(x + 1)(x – 1) 10. 3(x + 7)(x – 5) 11. 4(x + 5)(x – 5) 12. 3(x2 + 9) 13. 6x(x + 1)(x – 1) 14. x(x – 8)(x – 6) 15. x2 (x – 7)(x + 1) 16. x(x + 6)(x – 6) 17. 4(x2 – 2x + 7) 18. 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. 1, 4 14. -4, 7 15. -4, 4 16. -3, 7 17. -8, 2 18. -1, 6 19. 9, -2 20. -1, 9 21. -2, 7

22. -1,

23. , 3

24. -3,

25. - , -2

26. -11, 1 27. 3, 6 28. -10, 5 29. 6, 5

30. -4,

31. -4, 12

Worksheet 46

1. 18 tablets 2. $2475 3. 32.5 ft. 4. 15 pages 5. freight: 35 mph express: 55 mph 6. 200 mph 7. 35 mph 8. h: 180 p: 380 9. 15 hrs 10. 24 min 11. 6 min 12. 10 min

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. x4

2. x5

3.

4.

5. x3 y3

6. x5 y2

7. 3x3

8. 4 y5 9. 6x

10.

11.

12.

13.

14.

15.

16.

17. 6ab3

18. 10x2y8

19.

20.

21.

22.

23. x2y2

24.

25.

26.

27.

28. 10(a + 5) 2

29. 8(x + y) 2 30. 7(x - 3) 31. 2(x - 1) 32. x + 6 33. b - 5

Worksheet 51

1. 3

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. =

12. =

13. =

14. =

=

15. =

Worksheet 52 1. 9i 2. 7i 3.

4.

5. 6. i 7.

8.

9.

10.

Worksheet 53

1. 5 2. 15 3. 5 4. 5. 6. 7. 8. 100 9. 10.

Worksheet 54

1. -2, 2 2. -4, 4 3. -9, 9 4. -12.12

5. ,

6. ,

7. ,

8. ,

9. ,

10. ,

11. ,

12. ,

13. No real number solution 14. No real number solution

15.

16.

17. 18. -4, 2 19. -4, 8 20. -2, -8 21. 6, -2

22. ,

23. ,

24. ,

25. ,

26.

27.

28.

29.

Worksheet 55 1. a = +1; b = -1; c = -42 2. a = +1; b = +8; c = -20 3. a = +1; b = -10; c = -24 4. a = +2; b = +3; c = +6 5. a = +2; b = -11; c = +10

6.

7.

8.

9.

10.

11. x = +7; x = -6 12. x = -10; x = +2 13. y = -2; y = +12

14.

15.