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Intermediate Algebra 2e

Review Exercises

Intermediate Algebra 2eReview Exercises
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

Add and Subtract Polynomials

Types of Polynomials

In the following exercises, determine the type of polynomial.

342.

16 x 2 − 40 x − 25 16 x 2 − 40 x − 25

343.

5 m + 9 5 m + 9

344.

−15 −15

345.

y 2 + 6 y 3 + 9 y 4 y 2 + 6 y 3 + 9 y 4

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

346.

4 p + 11 p 4 p + 11 p

347.

−8 y 3 − 5 y 3 −8 y 3 − 5 y 3

348.

( 4 a 2 + 9 a − 11 ) + ( 6 a 2 − 5 a + 10 ) ( 4 a 2 + 9 a − 11 ) + ( 6 a 2 − 5 a + 10 )

349.

( 8 m 2 + 12 m − 5 ) − ( 2 m 2 − 7 m − 1 ) ( 8 m 2 + 12 m − 5 ) − ( 2 m 2 − 7 m − 1 )

350.

( y 2 − 3 y + 12 ) + ( 5 y 2 − 9 ) ( y 2 − 3 y + 12 ) + ( 5 y 2 − 9 )

351.

( 5 u 2 + 8 u ) − ( 4 u − 7 ) ( 5 u 2 + 8 u ) − ( 4 u − 7 )

352.

Find the sum of 8q3−278q3−27 and q2+6q−2.q2+6q−2.

353.

Find the difference of x2+6x+8x2+6x+8 and x2−8x+15.x2−8x+15.

In the following exercises, simplify.

354.

17 m n 2 − ( −9 m n 2 ) + 3 m n 2 17 m n 2 − ( −9 m n 2 ) + 3 m n 2

355.

18 a − 7 b − 21 a 18 a − 7 b − 21 a

356.

2 p q 2 − 5 p − 3 q 2 2 p q 2 − 5 p − 3 q 2

357.

( 6 a 2 + 7 ) + ( 2 a 2 − 5 a − 9 ) ( 6 a 2 + 7 ) + ( 2 a 2 − 5 a − 9 )

358.

( 3 p 2 − 4 p − 9 ) + ( 5 p 2 + 14 ) ( 3 p 2 − 4 p − 9 ) + ( 5 p 2 + 14 )

359.

( 7 m 2 − 2 m − 5 ) − ( 4 m 2 + m − 8 ) ( 7 m 2 − 2 m − 5 ) − ( 4 m 2 + m − 8 )

360.

( 7 b 2 − 4 b + 3 ) − ( 8 b 2 − 5 b − 7 ) ( 7 b 2 − 4 b + 3 ) − ( 8 b 2 − 5 b − 7 )

361.

Subtract (8y2−y+9)(8y2−y+9) from (11y2−9y−5)(11y2−9y−5)

362.

Find the difference of (z2−4z−12)(z2−4z−12) and (3z2+2z−11)(3z2+2z−11)

363.

( x 3 − x 2 y ) − ( 4 x y 2 − y 3 ) + ( 3 x 2 y − x y 2 ) ( x 3 − x 2 y ) − ( 4 x y 2 − y 3 ) + ( 3 x 2 y − x y 2 )

364.

( x 3 − 2 x 2 y ) − ( x y 2 − 3 y 3 ) − ( x 2 y − 4 x y 2 ) ( x 3 − 2 x 2 y ) − ( x y 2 − 3 y 3 ) − ( x 2 y − 4 x y 2 )

Evaluate a Polynomial Function for a Given Value of the Variable

In the following exercises, find the function values for each polynomial function.

365.

For the function f(x)=7x2−3x+5f(x)=7x2−3x+5 find:
ⓐ f(5)f(5) ⓑ f(−2)f(−2) ⓒ f(0)f(0)

366.

For the function g(x)=15−16x2,g(x)=15−16x2, find:
ⓐ g(−1)g(−1) ⓑ g(0)g(0) ⓒ g(2)g(2)

367.

A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function h(t)=−16t2+640h(t)=−16t2+640 gives the height of the glasses t seconds after they were dropped. Find the height of the glasses when t=6.t=6.

368.

A manufacturer of the latest soccer shoes has found that the revenue received from selling the shoes at a cost of pp dollars each is given by the polynomial R(p)=−5p2+360p.R(p)=−5p2+360p. Find the revenue received when p=10p=10 dollars.

Add and Subtract Polynomial Functions

In the following exercises, find ⓐ (f + g)(x) ⓑ (f + g)(3) ⓒ (f − g)(x) ⓓ (f − g)(−2)

369.

f(x)=2x2−4x−7f(x)=2x2−4x−7 and g(x)=2x2−x+5g(x)=2x2−x+5

370.

f(x)=4x3−3x2+x−1f(x)=4x3−3x2+x−1 and g(x)=8x3−1g(x)=8x3−1

Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

371.

p 3 · p 10 p 3 · p 10

372.

2 · 2 6 2 · 2 6

373.

a · a 2 · a 3 a · a 2 · a 3

374.

x · x 8 x · x 8

375.

y a · y b y a · y b

376.

2 8 2 2 2 8 2 2

377.

a 6 a a 6 a

378.

n 3 n 12 n 3 n 12

379.

1 x 5 1 x 5

380.

3 0 3 0

381.

y 0 y 0

382.

( 14 t ) 0 ( 14 t ) 0

383.

12 a 0 − 15 b 0 12 a 0 − 15 b 0

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

384.

6 −2 6 −2

385.

( −10 ) −3 ( −10 ) −3

386.

5 · 2 −4 5 · 2 −4

387.

( 8 n ) −1 ( 8 n ) −1

388.

y −5 y −5

389.

10 −3 10 −3

390.

1 a −4 1 a −4

391.

1 6 −2 1 6 −2

392.

− 5 −3 − 5 −3

393.

( − 1 5 ) −3 ( − 1 5 ) −3

394.

− ( 1 2 ) −3 − ( 1 2 ) −3

395.

( −5 ) −3 ( −5 ) −3

396.

( 5 9 ) −2 ( 5 9 ) −2

397.

( − 3 x ) −3 ( − 3 x ) −3

In the following exercises, simplify each expression using the Product Property.

398.

( y 4 ) 3 ( y 4 ) 3

399.

( 3 2 ) 5 ( 3 2 ) 5

400.

( a 10 ) y ( a 10 ) y

401.

x −3 · x 9 x −3 · x 9

402.

r −5 · r −4 r −5 · r −4

403.

( u v −3 ) ( u −4 v −2 ) ( u v −3 ) ( u −4 v −2 )

404.

( m 5 ) −1 ( m 5 ) −1

405.

p 5 · p −2 · p −4 p 5 · p −2 · p −4

In the following exercises, simplify each expression using the Power Property.

406.

( k −2 ) −3 ( k −2 ) −3

407.

q 4 q 20 q 4 q 20

408.

b 8 b −2 b 8 b −2

409.

n −3 n −5 n −3 n −5

In the following exercises, simplify each expression using the Product to a Power Property.

410.

( −5 a b ) 3 ( −5 a b ) 3

411.

( −4 p q ) 0 ( −4 p q ) 0

412.

( −6 x 3 ) −2 ( −6 x 3 ) −2

413.

( 3 y −4 ) 2 ( 3 y −4 ) 2

In the following exercises, simplify each expression using the Quotient to a Power Property.

414.

( 3 5 x ) −2 ( 3 5 x ) −2

415.

( 3 x y 2 z ) 4 ( 3 x y 2 z ) 4

416.

( 4 p −3 q 2 ) 2 ( 4 p −3 q 2 ) 2

In the following exercises, simplify each expression by applying several properties.

417.

( x 2 y ) 2 ( 3 x y 5 ) 3 ( x 2 y ) 2 ( 3 x y 5 ) 3

418.

( −3 a −2 ) 4 ( 2 a 4 ) 2 ( −6 a 2 ) 3 ( −3 a −2 ) 4 ( 2 a 4 ) 2 ( −6 a 2 ) 3

419.

( 3 x y 3 4 x 4 y −2 ) 2 ( 6 x y 4 8 x 3 y −2 ) −1 ( 3 x y 3 4 x 4 y −2 ) 2 ( 6 x y 4 8 x 3 y −2 ) −1

In the following exercises, write each number in scientific notation.

420.

2.568 2.568

421.

5,300,000

422.

0.00814 0.00814

In the following exercises, convert each number to decimal form.

423.

2.9 × 10 4 2.9 × 10 4

424.

3.75 × 10 −1 3.75 × 10 −1

425.

9.413 × 10 −5 9.413 × 10 −5

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

426.

( 3 × 10 7 ) ( 2 × 10 −4 ) ( 3 × 10 7 ) ( 2 × 10 −4 )

427.

( 1.5 × 10 −3 ) ( 4.8 × 10 −1 ) ( 1.5 × 10 −3 ) ( 4.8 × 10 −1 )

428.

6 × 10 9 2 × 10 −1 6 × 10 9 2 × 10 −1

429.

9 × 10 −3 1 × 10 −6 9 × 10 −3 1 × 10 −6

Multiply Polynomials

Multiply Monomials

In the following exercises, multiply the monomials.

430.

( −6 p 4 ) ( 9 p ) ( −6 p 4 ) ( 9 p )

431.

( 1 3 c 2 ) ( 30 c 8 ) ( 1 3 c 2 ) ( 30 c 8 )

432.

( 8 x 2 y 5 ) ( 7 x y 6 ) ( 8 x 2 y 5 ) ( 7 x y 6 )

433.

( 2 3 m 3 n 6 ) ( 1 6 m 4 n 4 ) ( 2 3 m 3 n 6 ) ( 1 6 m 4 n 4 )

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

434.

7 ( 10 − x ) 7 ( 10 − x )

435.

a 2 ( a 2 − 9 a − 36 ) a 2 ( a 2 − 9 a − 36 )

436.

−5 y ( 125 y 3 − 1 ) −5 y ( 125 y 3 − 1 )

437.

( 4 n − 5 ) ( 2 n 3 ) ( 4 n − 5 ) ( 2 n 3 )

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using:

ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

438.

( a + 5 ) ( a + 2 ) ( a + 5 ) ( a + 2 )

439.

( y − 4 ) ( y + 12 ) ( y − 4 ) ( y + 12 )

440.

( 3 x + 1 ) ( 2 x − 7 ) ( 3 x + 1 ) ( 2 x − 7 )

441.

( 6 p − 11 ) ( 3 p − 10 ) ( 6 p − 11 ) ( 3 p − 10 )

In the following exercises, multiply the binomials. Use any method.

442.

( n + 8 ) ( n + 1 ) ( n + 8 ) ( n + 1 )

443.

( k + 6 ) ( k − 9 ) ( k + 6 ) ( k − 9 )

444.

( 5 u − 3 ) ( u + 8 ) ( 5 u − 3 ) ( u + 8 )

445.

( 2 y − 9 ) ( 5 y − 7 ) ( 2 y − 9 ) ( 5 y − 7 )

446.

( p + 4 ) ( p + 7 ) ( p + 4 ) ( p + 7 )

447.

( x − 8 ) ( x + 9 ) ( x − 8 ) ( x + 9 )

448.

( 3 c + 1 ) ( 9 c − 4 ) ( 3 c + 1 ) ( 9 c − 4 )

449.

( 10 a − 1 ) ( 3 a − 3 ) ( 10 a − 1 ) ( 3 a − 3 )

Multiply a Polynomial by a Polynomial

In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

450.

( x + 1 ) ( x 2 − 3 x − 21 ) ( x + 1 ) ( x 2 − 3 x − 21 )

451.

( 5 b − 2 ) ( 3 b 2 + b − 9 ) ( 5 b − 2 ) ( 3 b 2 + b − 9 )

In the following exercises, multiply. Use either method.

452.

( m + 6 ) ( m 2 − 7 m − 30 ) ( m + 6 ) ( m 2 − 7 m − 30 )

453.

( 4 y − 1 ) ( 6 y 2 − 12 y + 5 ) ( 4 y − 1 ) ( 6 y 2 − 12 y + 5 )

Multiply Special Products

In the following exercises, square each binomial using the Binomial Squares Pattern.

454.

( 2 x − y ) 2 ( 2 x − y ) 2

455.

( x + 3 4 ) 2 ( x + 3 4 ) 2

456.

( 8 p 3 − 3 ) 2 ( 8 p 3 − 3 ) 2

457.

( 5 p + 7 q ) 2 ( 5 p + 7 q ) 2

In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

458.

( 3 y + 5 ) ( 3 y − 5 ) ( 3 y + 5 ) ( 3 y − 5 )

459.

( 6 x + y ) ( 6 x − y ) ( 6 x + y ) ( 6 x − y )

460.

( a + 2 3 b ) ( a − 2 3 b ) ( a + 2 3 b ) ( a − 2 3 b )

461.

( 12 x 3 − 7 y 2 ) ( 12 x 3 + 7 y 2 ) ( 12 x 3 − 7 y 2 ) ( 12 x 3 + 7 y 2 )

462.

( 13 a 2 − 8 b 4 ) ( 13 a 2 + 8 b 4 ) ( 13 a 2 − 8 b 4 ) ( 13 a 2 + 8 b 4 )

Divide Monomials

Divide Monomials

In the following exercises, divide the monomials.

463.

72 p 12 ÷ 8 p 3 72 p 12 ÷ 8 p 3

464.

−26 a 8 ÷ ( 2 a 2 ) −26 a 8 ÷ ( 2 a 2 )

465.

45 y 6 −15 y 10 45 y 6 −15 y 10

466.

−30 x 8 −36 x 9 −30 x 8 −36 x 9

467.

28 a 9 b 7 a 4 b 3 28 a 9 b 7 a 4 b 3

468.

11 u 6 v 3 55 u 2 v 8 11 u 6 v 3 55 u 2 v 8

469.

( 5 m 9 n 3 ) ( 8 m 3 n 2 ) ( 10 m n 4 ) ( m 2 n 5 ) ( 5 m 9 n 3 ) ( 8 m 3 n 2 ) ( 10 m n 4 ) ( m 2 n 5 )

470.

( 42 r 2 s 4 ) ( 54 r s 2 ) ( 6 r s 3 ) ( 9 s ) ( 42 r 2 s 4 ) ( 54 r s 2 ) ( 6 r s 3 ) ( 9 s )

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial

471.

( 54 y 4 − 24 y 3 ) ÷ ( −6 y 2 ) ( 54 y 4 − 24 y 3 ) ÷ ( −6 y 2 )

472.

63 x 3 y 2 − 99 x 2 y 3 − 45 x 4 y 3 9 x 2 y 2 63 x 3 y 2 − 99 x 2 y 3 − 45 x 4 y 3 9 x 2 y 2

473.

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

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

474.

( 4 x 2 − 21 x − 18 ) ÷ ( x − 6 ) ( 4 x 2 − 21 x − 18 ) ÷ ( x − 6 )

475.

( y 2 + 2 y + 18 ) ÷ ( y + 5 ) ( y 2 + 2 y + 18 ) ÷ ( y + 5 )

476.

( n 3 − 2 n 2 − 6 n + 27 ) ÷ ( n + 3 ) ( n 3 − 2 n 2 − 6 n + 27 ) ÷ ( n + 3 )

477.

( a 3 − 1 ) ÷ ( a + 1 ) ( a 3 − 1 ) ÷ ( a + 1 )

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

478.

x3−3x2−4x+12x3−3x2−4x+12 is divided by x+2x+2

479.

2x3−11x2+11x+122x3−11x2+11x+12 is divided by x−3x−3

480.

x4+x2+6x−10x4+x2+6x−10 is divided by x+2x+2

Divide Polynomial Functions

In the following exercises, divide.

481.

For functions f(x)=x2−15x+54f(x)=x2−15x+54 and g(x)=x−9,g(x)=x−9, find ⓐ (fg)(x)(fg)(x)
ⓑ (fg)(−2)(fg)(−2)

482.

For functions f(x)=x3+x2−7x+2f(x)=x3+x2−7x+2 and g(x)=x−2,g(x)=x−2, find ⓐ (fg)(x)(fg)(x)
ⓑ (fg)(3)(fg)(3)

Use the Remainder and Factor Theorem

In the following exercises, use the Remainder Theorem to find the remainder.

483.

f(x)=x3−4x−9f(x)=x3−4x−9 is divided by x+2x+2

484.

f(x)=2x3−6x−24f(x)=2x3−6x−24 divided by x−3x−3

In the following exercises, use the Factor Theorem to determine if x−cx−c is a factor of the polynomial function.

485.

Determine whether x−2x−2 is a factor of x3−7x2+7x−6x3−7x2+7x−6.

486.

Determine whether x−3x−3 is a factor of x3−7x2+11x+3x3−7x2+11x+3.

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