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Intermediate Algebra 2e

Review Exercises

Intermediate Algebra 2eReview Exercises
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Table of contents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    11. Chapter 11
    12. Chapter 12
  15. Index

Review Exercises

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.

337.

12 a 2 b 3 , 15 a b 2 12 a 2 b 3 , 15 a b 2

338.

12 m 2 n 3 , 42 m 5 n 3 12 m 2 n 3 , 42 m 5 n 3

339.

15 y 3 , 21 y 2 , 30 y 15 y 3 , 21 y 2 , 30 y

340.

45 x 3 y 2 , 15 x 4 y , 10 x 5 y 3 45 x 3 y 2 , 15 x 4 y , 10 x 5 y 3

Factor the Greatest Common Factor from a Polynomial

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

341.

35 y + 84 35 y + 84

342.

6 y 2 + 12 y − 6 6 y 2 + 12 y − 6

343.

18 x 3 − 15 x 18 x 3 − 15 x

344.

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

345.

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

346.

−3 x + 24 −3 x + 24

347.

−3 x 3 + 27 x 2 − 12 x −3 x 3 + 27 x 2 − 12 x

348.

3 x ( x − 1 ) + 5 ( x − 1 ) 3 x ( x − 1 ) + 5 ( x − 1 )

Factor by Grouping

In the following exercises, factor by grouping.

349.

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

350.

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

351.

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

352.

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

353.

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

354.

5 x − 5 y − y + x 5 x − 5 y − y + x

Factor Trinomials

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

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

355.

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

356.

k 2 − 16 k + 60 k 2 − 16 k + 60

357.

m 2 + 3 m − 54 m 2 + 3 m − 54

358.

x 2 − 3 x − 10 x 2 − 3 x − 10

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

359.

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

360.

r 2 + 3 r s − 28 s 2 r 2 + 3 r s − 28 s 2

361.

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

362.

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

363.

m 2 − 5 m n + 30 n 2 m 2 − 5 m n + 30 n 2

Factor Trinomials of the Form ax2+bx+cax2+bx+c Using Trial and Error

In the following exercises, factor completely using trial and error.

364.

x 3 + 5 x 2 − 24 x x 3 + 5 x 2 − 24 x

365.

3 y 3 − 21 y 2 + 30 y 3 y 3 − 21 y 2 + 30 y

366.

5 x 4 + 10 x 3 − 75 x 2 5 x 4 + 10 x 3 − 75 x 2

367.

5 y 2 + 14 y + 9 5 y 2 + 14 y + 9

368.

8 x 2 + 25 x + 3 8 x 2 + 25 x + 3

369.

10 y 2 − 53 y − 11 10 y 2 − 53 y − 11

370.

6 p 2 − 19 p q + 10 q 2 6 p 2 − 19 p q + 10 q 2

371.

−81 a 2 + 153 a + 18 −81 a 2 + 153 a + 18

Factor Trinomials of the Form ax2+bx+cax2+bx+c using the ‘ac’ Method

In the following exercises, factor.

372.

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

373.

18 a 2 − 9 a + 1 18 a 2 − 9 a + 1

374.

15 p 2 + 2 p − 8 15 p 2 + 2 p − 8

375.

15 x 2 + 6 x − 2 15 x 2 + 6 x − 2

376.

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

377.

3 x 2 + 3 x − 36 3 x 2 + 3 x − 36

378.

48 y 2 + 12 y − 36 48 y 2 + 12 y − 36

379.

18 a 2 − 57 a − 21 18 a 2 − 57 a − 21

380.

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

Factor using substitution

In the following exercises, factor using substitution.

381.

x 4 − 13 x 2 − 30 x 4 − 13 x 2 − 30

382.

( x − 3 ) 2 − 5 ( x − 3 ) − 36 ( x − 3 ) 2 − 5 ( x − 3 ) − 36

Factor Special Products

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

383.

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

384.

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

385.

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

386.

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

387.

75 u 4 − 30 u 3 v + 3 u 2 v 2 75 u 4 − 30 u 3 v + 3 u 2 v 2

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

388.

81 r 2 − 25 81 r 2 − 25

389.

169 m 2 − n 2 169 m 2 − n 2

390.

25 p 2 − 1 25 p 2 − 1

391.

9 − 121 y 2 9 − 121 y 2

392.

20 x 2 − 125 20 x 2 − 125

393.

169 n 3 − n 169 n 3 − n

394.

6 p 2 q 2 − 54 p 2 6 p 2 q 2 − 54 p 2

395.

24 p 2 + 54 24 p 2 + 54

396.

49 x 2 − 81 y 2 49 x 2 − 81 y 2

397.

16 z 4 − 1 16 z 4 − 1

398.

48 m 4 n 2 − 243 n 2 48 m 4 n 2 − 243 n 2

399.

a 2 + 6 a + 9 − 9 b 2 a 2 + 6 a + 9 − 9 b 2

400.

x 2 − 16 x + 64 − y 2 x 2 − 16 x + 64 − y 2

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

401.

a 3 − 125 a 3 − 125

402.

b 3 − 216 b 3 − 216

403.

2 m 3 + 54 2 m 3 + 54

404.

81 m 3 + 3 81 m 3 + 3

General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

405.

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

406.

24 a 4 − 9 a 3 24 a 4 − 9 a 3

407.

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

408.

6 a 2 − 25 a − 9 6 a 2 − 25 a − 9

409.

5 u 4 − 45 u 2 5 u 4 − 45 u 2

410.

n 4 − 81 n 4 − 81

411.

64 j 2 + 225 64 j 2 + 225

412.

5 x 2 + 5 x − 60 5 x 2 + 5 x − 60

413.

b 3 − 64 b 3 − 64

414.

m 3 + 125 m 3 + 125

415.

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

416.

48 x 5 y 2 − 243 x y 2 48 x 5 y 2 − 243 x y 2

417.

5 q 2 − 15 q − 90 5 q 2 − 15 q − 90

418.

4 u 5 v + 4 u 2 v 3 4 u 5 v + 4 u 2 v 3

419.

10 m 4 − 6250 10 m 4 − 6250

420.

60 x 2 y − 75 x y + 30 y 60 x 2 y − 75 x y + 30 y

421.

16 x 2 − 24 x y + 9 y 2 − 64 16 x 2 − 24 x y + 9 y 2 − 64

Polynomial Equations

Use the Zero Product Property

In the following exercises, solve.

422.

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

423.

( 5 b + 1 ) ( 6 b + 1 ) = 0 ( 5 b + 1 ) ( 6 b + 1 ) = 0

424.

6 m ( 12 m − 5 ) = 0 6 m ( 12 m − 5 ) = 0

425.

( 2 x − 1 ) 2 = 0 ( 2 x − 1 ) 2 = 0

426.

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

Solve Quadratic Equations by Factoring

In the following exercises, solve.

427.

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

428.

y 2 − y − 72 = 0 y 2 − y − 72 = 0

429.

2 p 2 − 11 p = 40 2 p 2 − 11 p = 40

430.

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

431.

144 m 2 − 25 = 0 144 m 2 − 25 = 0

432.

4 n 2 = 36 4 n 2 = 36

433.

( x + 6 ) ( x − 3 ) = −8 ( x + 6 ) ( x − 3 ) = −8

434.

( 3 x − 2 ) ( x + 4 ) = 12 x ( 3 x − 2 ) ( x + 4 ) = 12 x

435.

16 p 3 = 24 p 2 − 9 p 16 p 3 = 24 p 2 − 9 p

436.

2 y 3 + 2 y 2 = 12 y 2 y 3 + 2 y 2 = 12 y

Solve Equations with Polynomial Functions

In the following exercises, solve.

437.

For the function, f(x)=x2+11x+20,f(x)=x2+11x+20, ⓐ find when f(x)=−8f(x)=−8 ⓑ Use this information to find two points that lie on the graph of the function.

438.

For the function, f(x)=9x2−18x+5,f(x)=9x2−18x+5, ⓐ find when f(x)=−3f(x)=−3 ⓑ Use this information to find two points that lie on the graph of the function.

In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

439.

f ( x ) = 64 x 2 − 49 f ( x ) = 64 x 2 − 49

440.

f ( x ) = 6 x 2 − 13 x − 5 f ( x ) = 6 x 2 − 13 x − 5

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

441.

The product of two consecutive odd numbers is 399. Find the numbers.

442.

The area of a rectangular shaped patio 432 square feet. The length of the patio is 6 feet more than its width. Find the length and width.

443.

A ladder leans against the wall of a building. The length of the ladder is 9 feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is 7 feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning against the building.

444.

Shruti is going to throw a ball from the top of a cliff. When she throws the ball from 80 feet above the ground, the function h(t)=−16t2+64t+80h(t)=−16t2+64t+80 models the height, h, of the ball above the ground as a function of time, t. Find: ⓐ the zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 80 feet above the ground. ⓒ the height the ball will be at t=2t=2 seconds which is when the ball will be at its highest point.

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