 Intermediate Algebra

# Review Exercises

Intermediate AlgebraReview Exercises

### Review Exercises

Determine the Degree 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.

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

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 $p=110p=110$ dollars.

In the following exercises, find (f + g)(x)  (f + g)(3)  (fg)(x)  (fg)(−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 $f(x)=x2−15x+45f(x)=x2−15x+45$ 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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