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Mathematics LibreTexts

5.E: Review Exercises and Sample Exam

  • Anonymous
  • LibreTexts

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Review Exercises

Exercise 5.E.1 Rules of Exponents

Simplify.

  1. 7376
  2. 5956
  3. y5y2y3
  4. x3y2xy3
  5. 5a3b2c6a2bc2
  6. 55x2yz55xyz2
  7. (3a2b42c3)2
  8. (2a3b4c4)3
  9. 5x3y0(z2)32x4(y3)2z
  10. (25x6y5z)0
  11. Each side of a square measures 5x2 units. Find the area of the square in terms of x.
  12. Each side of a cube measures 2x3 units. Find the volume of the cube in terms of x.
Answer

1. 79

3. y10

5. 30a5b3c3

7. 9a4b84c6

9. 10x7y6z7

11. A=25x4

Exercise 5.E.2 Introduction to Polynomials

Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.

  1. 8a31
  2. 5y2y+1
  3. 12ab2
  4. 10
Answer

1. Binomial; degree 3

3. Monomial; degree 3

Exercise 5.E.3 Introduction to Polynomials

Write the following polynomials in standard form.

  1. 7x25x
  2. 5x213x+2x3
Answer

1. x25x+7

Exercise 5.E.4 Introduction to Polynomials

Evaluate.

  1. 2x2x+1, where x=3
  2. 12x34, where x=13
  3. b24ac, where a=12,b=3, and c=32
  4. a2b2, where a=12 and b=13
  5. a3b3, where a=2 and b=1
  6. xy22x2y, where x=3 and y=1
  7. Given f(x)=3x25x+2, find f(2).
  8. Given g(x)=x3x2+x1, find g(1).
  9. The surface area of a rectangular solid is given by the formula SA=2lw+2wh+2lh, where l,w, and h represent the length, width, and height, respectively. If the length of a rectangular solid measures 2 units, the width measures 3 units, and the height measures 5 units, then calculate the surface area.
  10. The surface area of a sphere is given by the formula SA=4πr2, where r represents the radius of the sphere. If a sphere has a radius of 5 units, then calculate the surface area.
Answer

1. 22

3. 6

5. 7

7. f(2)=24

9. 62 square units

Exercise 5.E.5 Adding and Subtracting Polynomials

Perform the operations.

  1. (3x4)+(9x1)
  2. (13x19)+(16x+12)
  3. (7x2x+9)+(x25x+6)
  4. (6x2y5xy23)+(2x2y+3xy2+1)
  5. (4y+7)(6y2)+(10y1)
  6. (5y23y+1)(8y2+6y11)
  7. (7x2y23xy+6)(6x2y2+2xy1)
  8. (a3b3)(a3+1)(b31)
  9. (x5x3+x1)(x4x2+5)
  10. (5x34x2+x3)(5x33)+(4x2x)
  11. Subtract 2x1 from 9x+8.
  12. Subtract 3x210x2 from 5x2+x5.
  13. Given f(x)=3x2x+5 and g(x)=x29, find (f+g)(x).
  14. Given f(x)=3x2x+5 and g(x)=x29, find (fg)(x).
  15. Given f(x)=3x2x+5 and g(x)=x29, find (f+g)(2).
  16. Given f(x)=3x2x+5 and g(x)=x29, find (fg)(2).
Answer

1. 12x5

3. 8x26x+15

5. 8y+8

7. x2y25xy+7

9. x5x4x3+x2+x6

11. 7x+9

13. (f+g)(x)=4x2x4

15. (f+g)(2)=14

Exercise 5.E.6 Multiplying Polynomials

Multiply.

  1. 6x2(5x4)
  2. 3ab2(7a2b)
  3. 2y(5y12)
  4. 3x(3x2x+2)
  5. x2y(2x2y5xy2+2)
  6. 4ab(a28ab+b2)
  7. (x8)(x+5)
  8. (2y5)(2y+5)
  9. (3x1)2
  10. (3x1)3
  11. (2x1)(5x23x+1)
  12. (x2+3)(x32x1)
  13. (5y+7)2
  14. (y21)2
  15. Find the product of x21 and x2+1.
  16. Find the product of 32x2y and 10x30y+2.
  17. Given f(x)=7x2 and g(x)=x23x+1, find (fg)(x).
  18. Given f(x)=x5 and g(x)=x29, find (fg)(x).
  19. Given f(x)=7x2 and g(x)=x23x+1, find (fg)(1).
  20. Given f(x)=x5 and g(x)=x29, find (fg)(1).
Answer

1. 30x6

3. 10y224y

5. 2x4y25x3y3+2x2y

7. x23x40

9. 9x26x+1

11. 10x311x2+5x1

13. 25y2+70y+49

15. x41

17. (fg)(x)=7x323x2+13x2

19. (fg)(1)=45

Exercise 5.E.7 Dividing Polynomials

Divide.

  1. 7y214y+287
  2. 12x530x3+6x6x
  3. 4a2b16ab24ab4ab
  4. 6a624a4+5a23a2
  5. (10x219x+6)÷(2x3)
  6. (2x35x2+5x6)÷(x2)
  7. 10x421x316x2+23x202x5
  8. x53x428x3+61x212x+36x6
  9. 10x355x2+72x42x7
  10. 3x4+19x3+3x216x113x+1
  11. 5x4+4x35x2+21x+215x+4
  12. x44x4
  13. 2x4+10x323x215x+302x23
  14. 7x417x3+17x211x+2x22x+1
  15. Given f(x)=x34x+1 and g(x)=x1, find (f/g)(x).
  16. Given f(x)=x532 and g(x)=x2, find (f/g)(x).
  17. Given f(x)=x34x+1 and g(x)=x1, find (f/g)(2).
  18. Given f(x)=x532 and g(x)=x2, find (f/g)(0).
Answer

1. y22y+4

3. a+4b+1

5. 5x2

7. 5x3+2x23x+4

9. 5x210x+1+32x7

11. x3x+5+15x+4

13. x2+5x10

15. (f/g)(x)=x2+x32x1

17. (f/g)(2)=1

Exercise 5.E.8 Negative Exponents

Simplify.

  1. (10)2
  2. 102
  3. 5x3
  4. (5x)3
  5. 17y3
  6. 3x4y2
  7. 2a2b5c8
  8. (5x2yz1)2
  9. (2x3y0z2)3
  10. (10a5b3c25ab2c2)1
  11. (a2b4c02a4b3c)3
Answer

1. 1100

3. 5x3

5. y37

7. 2a2c8b5

9. x98z6

11. 8a6b3c3

Exercise 5.E.9 Negative Exponents

The value in dollars of a new laptop computer can be estimated by using the formula V=1200(t+1)1, where t represents the number of years after the purchase.

  1. Estimate the value of the laptop when it is 112 years old.
  2. What was the laptop worth new?
Answer

2. $1,200

Exercise 5.E.10 Negative Exponents

Rewrite using scientific notation.

  1. 2,030,000,000
  2. 0.00000004011
Answer

2. 5.796×1019

Exercise 5.E.11 Negative Exponents

Perform the indicated operations.

  1. (5.2×1012)(1.8×103)
  2. (9.2×104)(6.3×1022)
  3. 4×10168×107
  4. 9×10304×1010
  5. 5,000,000,000,000×0.0000023
  6. 0.0003120,000,000,000,000
Answer

2. 5.796×1019

4. 2.25×1020

6. 2.5×1018

Simple Exam

Exercise 5.E.12

Simplify.

  1. 5x3(2x2y)
  2. (x2)4x3x
  3. (2x2y3)2x2y
    1. (5)0
    2. 50
Answer

1. 10x5y

3. 4x2y5

Exercise 5.E.13

Evaluate.

  1. 2x2x+5, where x=5
  2. a2b2, where a=4 and b=3
Answer

1. 60

Exercise 5.E.14

Perform the operations.

  1. (3x24x+5)+(7x2+9x2)
  2. (8x25x+1)(10x2+2x1)
  3. (35a12)(23a2+23a29)+(115a518)
  4. 2x2(2x33x24x+5)
  5. (2x3)(x+5)
  6. (x1)3
  7. 81x5y2z3x3yz
  8. 10x915x5+5x25x2
  9. x35x2+7x2x2
  10. 6x4x313x22x12x1
Answer

1. 4x2+5x+3

3. 23a259

5. 2x2+7x15

7. 27x2y

9. x23x+1

Exercise 5.E.15

Simplify.

  1. 23
  2. 5x2
  3. (2x4y3z)2
  4. (2a3b5c2ab3c2)3
  5. Subtract 5x2y4xy2+1 from 10x2y6xy2+2.
  6. If each side of a cube measures 4x4 units, calculate the volume in terms of x.
  7. The height of a projectile in feet is given by the formula h=16t2+96t+10, where t represents time in seconds. Calculate the height of the projectile at 112 seconds.
  8. The cost in dollars of producing custom t-shirts is given by the formula C=120+3.50x, where x represents the number of t-shirts produced. The revenue generated by selling the t-shirts for $6.50 each is given by the formula R=6.50x, where x represents the number of t-shirts sold.
    1. Find a formula for the profit. (profit = revenue − cost)
    2. Use the formula to calculate the profit from producing and selling 150 t-shirts.
  9. The total volume of water in earth’s oceans, seas, and bays is estimated to be 4.73×1019 cubic feet. By what factor is the volume of the moon, 7.76×1020 cubic feet, larger than the volume of earth’s oceans? Round to the nearest tenth.
Answer

1. 18

3. y64x8z2

5. 5x2y2xy2+1

7. 118 feet

9. 16.4


This page titled 5.E: Review Exercises and Sample Exam is shared under a not declared license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform.

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