Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Elementary Algebra 2e

Review Exercises

Elementary Algebra 2eReview Exercises
Menu
Table of contents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Chapter Review
      1. Key Terms
      2. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Chapter Review
      1. Key Terms
      2. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Review Exercises

7.1 Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

363.

42 , 60 42 , 60

364.

450 , 420 450 , 420

365.

90 , 150 , 105 90 , 150 , 105

366.

60 , 294 , 630 60 , 294 , 630

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

367.

24 x − 42 24 x − 42

368.

35 y + 84 35 y + 84

369.

15 m 4 + 6 m 2 n 15 m 4 + 6 m 2 n

370.

24 p t 4 + 16 t 7 24 p t 4 + 16 t 7

Factor by Grouping

In the following exercises, factor by grouping.

371.

a x − a y + b x − b y a x − a y + b x − b y

372.

x 2 y − x y 2 + 2 x − 2 y x 2 y − x y 2 + 2 x − 2 y

373.

x 2 + 7 x − 3 x − 21 x 2 + 7 x − 3 x − 21

374.

4 x 2 − 16 x + 3 x − 12 4 x 2 − 16 x + 3 x − 12

375.

m 3 + m 2 + m + 1 m 3 + m 2 + m + 1

376.

5 x − 5 y − y + x 5 x − 5 y − y + x

7.2 Factor Trinomials of the form x2+bx+cx2+bx+c

Factor Trinomials of the Form x2+bx+cx2+bx+c

In the following exercises, factor each trinomial of the form x2+bx+cx2+bx+c.

377.

u 2 + 17 u + 72 u 2 + 17 u + 72

378.

a 2 + 14 a + 33 a 2 + 14 a + 33

379.

k 2 − 16 k + 60 k 2 − 16 k + 60

380.

r 2 − 11 r + 28 r 2 − 11 r + 28

381.

y 2 + 6 y − 7 y 2 + 6 y − 7

382.

m 2 + 3 m − 54 m 2 + 3 m − 54

383.

s 2 − 2 s − 8 s 2 − 2 s − 8

384.

x 2 − 3 x − 10 x 2 − 3 x − 10

Factor Trinomials of the Form x2+bxy+cy2x2+bxy+cy2

In the following examples, factor each trinomial of the form x2+bxy+cy2x2+bxy+cy2.

385.

x 2 + 12 x y + 35 y 2 x 2 + 12 x y + 35 y 2

386.

u 2 + 14 u v + 48 v 2 u 2 + 14 u v + 48 v 2

387.

a 2 + 4 a b − 21 b 2 a 2 + 4 a b − 21 b 2

388.

p 2 − 5 p q − 36 q 2 p 2 − 5 p q − 36 q 2

7.3 Factoring Trinomials of the form ax2+bx+cax2+bx+c

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

389.

y 2 − 17 y + 42 y 2 − 17 y + 42

390.

12 r 2 + 32 r + 5 12 r 2 + 32 r + 5

391.

8 a 3 + 72 a 8 a 3 + 72 a

392.

4 m − m n − 3 n + 12 4 m − m n − 3 n + 12

Factor Trinomials of the Form ax2+bx+cax2+bx+c with a GCF

In the following exercises, factor completely.

393.

6 x 2 + 42 x + 60 6 x 2 + 42 x + 60

394.

8 a 2 + 32 a + 24 8 a 2 + 32 a + 24

395.

3 n 4 − 12 n 3 − 96 n 2 3 n 4 − 12 n 3 − 96 n 2

396.

5 y 3 + 25 y 2 − 70 y 5 y 3 + 25 y 2 − 70 y

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

397.

2 x 2 + 9 x + 4 2 x 2 + 9 x + 4

398.

3 y 2 + 17 y + 10 3 y 2 + 17 y + 10

399.

18 a 2 − 9 a + 1 18 a 2 − 9 a + 1

400.

8 u 2 − 14 u + 3 8 u 2 − 14 u + 3

401.

15 p 2 + 2 p − 8 15 p 2 + 2 p − 8

402.

15 x 2 + x − 2 15 x 2 + x − 2

403.

40 s 2 − s − 6 40 s 2 − s − 6

404.

20 n 2 − 7 n − 3 20 n 2 − 7 n − 3

Factor Trinomials with a GCF Using the “ac” Method

In the following exercises, factor.

405.

3 x 2 + 3 x − 36 3 x 2 + 3 x − 36

406.

4 x 2 + 4 x − 8 4 x 2 + 4 x − 8

407.

60 y 2 − 85 y − 25 60 y 2 − 85 y − 25

408.

18 a 2 − 57 a − 21 18 a 2 − 57 a − 21

7.4 Factoring Special Products

Factor Perfect Square Trinomials

In the following exercises, factor.

409.

25 x 2 + 30 x + 9 25 x 2 + 30 x + 9

410.

16 y 2 + 72 y + 81 16 y 2 + 72 y + 81

411.

36 a 2 − 84 a b + 49 b 2 36 a 2 − 84 a b + 49 b 2

412.

64 r 2 − 176 r s + 121 s 2 64 r 2 − 176 r s + 121 s 2

413.

40 x 2 + 360 x + 810 40 x 2 + 360 x + 810

414.

75 u 2 + 180 u + 108 75 u 2 + 180 u + 108

415.

2 y 3 − 16 y 2 + 32 y 2 y 3 − 16 y 2 + 32 y

416.

5 k 3 − 70 k 2 + 245 k 5 k 3 − 70 k 2 + 245 k

Factor Differences of Squares

In the following exercises, factor.

417.

81 r 2 − 25 81 r 2 − 25

418.

49 a 2 − 144 49 a 2 − 144

419.

169 m 2 − n 2 169 m 2 − n 2

420.

64 x 2 − y 2 64 x 2 − y 2

421.

25 p 2 − 1 25 p 2 − 1

422.

1 − 16 s 2 1 − 16 s 2

423.

9 − 121 y 2 9 − 121 y 2

424.

100 k 2 − 81 100 k 2 − 81

425.

20 x 2 − 125 20 x 2 − 125

426.

18 y 2 − 98 18 y 2 − 98

427.

49 u 3 − 9 u 49 u 3 − 9 u

428.

169 n 3 − n 169 n 3 − n

Factor Sums and Differences of Cubes

In the following exercises, factor.

429.

a 3 − 125 a 3 − 125

430.

b 3 − 216 b 3 − 216

431.

2 m 3 + 54 2 m 3 + 54

432.

81 x 3 + 3 81 x 3 + 3

7.5 General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

433.

24 x 3 + 44 x 2 24 x 3 + 44 x 2

434.

24 a 4 − 9 a 3 24 a 4 − 9 a 3

435.

16 n 2 − 56 m n + 49 m 2 16 n 2 − 56 m n + 49 m 2

436.

6 a 2 − 25 a − 9 6 a 2 − 25 a − 9

437.

5 r 2 + 22 r − 48 5 r 2 + 22 r − 48

438.

5 u 4 − 45 u 2 5 u 4 − 45 u 2

439.

n 4 − 81 n 4 − 81

440.

64 j 2 + 225 64 j 2 + 225

441.

5 x 2 + 5 x − 60 5 x 2 + 5 x − 60

442.

b 3 − 64 b 3 − 64

443.

m 3 + 125 m 3 + 125

444.

2 b 2 − 2 b c + 5 c b − 5 c 2 2 b 2 − 2 b c + 5 c b − 5 c 2

7.6 Quadratic Equations

Use the Zero Product Property

In the following exercises, solve.

445.

( a − 3 ) ( a + 7 ) = 0 ( a − 3 ) ( a + 7 ) = 0

446.

( b − 3 ) ( b + 10 ) = 0 ( b − 3 ) ( b + 10 ) = 0

447.

3 m ( 2 m − 5 ) ( m + 6 ) = 0 3 m ( 2 m − 5 ) ( m + 6 ) = 0

448.

7 n ( 3 n + 8 ) ( n − 5 ) = 0 7 n ( 3 n + 8 ) ( n − 5 ) = 0

Solve Quadratic Equations by Factoring

In the following exercises, solve.

449.

x 2 + 9 x + 20 = 0 x 2 + 9 x + 20 = 0

450.

y 2 − y − 72 = 0 y 2 − y − 72 = 0

451.

2 p 2 − 11 p = 40 2 p 2 − 11 p = 40

452.

q 3 + 3 q 2 + 2 q = 0 q 3 + 3 q 2 + 2 q = 0

453.

144 m 2 − 25 = 0 144 m 2 − 25 = 0

454.

4 n 2 = 36 4 n 2 = 36

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

455.

The product of two consecutive numbers is 462462. Find the numbers.

456.

The area of a rectangular shaped patio 400400 square feet. The length of the patio is 99 feet more than its width. Find the length and width.

PreviousNext
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
Citation information

© Jul 7, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.