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Intermediate Algebra 2e

Review Exercises

Intermediate Algebra 2eReview Exercises

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 8q3278q327 and q2+6q2.q2+6q2.

353.

Find the difference of x2+6x+8x2+6x+8 and x28x+15.x28x+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 (8y2y+9)(8y2y+9) from (11y29y5)(11y29y5)

362.

Find the difference of (z24z12)(z24z12) and (3z2+2z11)(3z2+2z11)

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)=7x23x+5f(x)=7x23x+5 find:
f(5)f(5) f(−2)f(−2) f(0)f(0)

366.

For the function g(x)=1516x2,g(x)=1516x2, 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)  (fg)(x)  (fg)(−2)

369.

f(x)=2x24x7f(x)=2x24x7 and g(x)=2x2x+5g(x)=2x2x+5

370.

f(x)=4x33x2+x1f(x)=4x33x2+x1 and g(x)=8x31g(x)=8x31

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.

x33x24x+12x33x24x+12 is divided by x+2x+2

479.

2x311x2+11x+122x311x2+11x+12 is divided by x3x3

480.

x4+x2+6x10x4+x2+6x10 is divided by x+2x+2

Divide Polynomial Functions

In the following exercises, divide.

481.

For functions f(x)=x215x+54f(x)=x215x+54 and g(x)=x9,g(x)=x9, find (fg)(x)(fg)(x)
(fg)(−2)(fg)(−2)

482.

For functions f(x)=x3+x27x+2f(x)=x3+x27x+2 and g(x)=x2,g(x)=x2, 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)=x34x9f(x)=x34x9 is divided by x+2x+2

484.

f(x)=2x36x24f(x)=2x36x24 divided by x3x3

In the following exercises, use the Factor Theorem to determine if xcxc is a factor of the polynomial function.

485.

Determine whether x2x2 is a factor of x37x2+7x6x37x2+7x6.

486.

Determine whether x3x3 is a factor of x37x2+11x+3x37x2+11x+3.

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