Elementary Algebra

# Review Exercises

Elementary AlgebraReview Exercises

### Review Exercises

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

588.

$11c4−23c2+111c4−23c2+1$
$9p3+6p2−p−59p3+6p2−p−5$
$37x+51437x+514$
10
$2y−122y−12$

589.

$a2−b2a2−b2$
$24d324d3$
$x2+8x−10x2+8x−10$
$m2n2−2mn+6m2n2−2mn+6$
$7y3+y2−2y−47y3+y2−2y−4$

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

590.
1. $3x2+9x+103x2+9x+10$
2. $14a2bc14a2bc$
3. $6y+16y+1$
4. $n3−4n2+2n−8n3−4n2+2n−8$
5. $−19−19$
591.
1. $5p3−8p2+10p−45p3−8p2+10p−4$
2. $−20q4−20q4$
3. $x2+6x+12x2+6x+12$
4. $23r2s2−4rs+523r2s2−4rs+5$
5. 100

In the following exercises, add or subtract the monomials.

592.

$5y 3 + 8 y 3 5y 3 + 8 y 3$

593.

$−14 k + 19 k −14 k + 19 k$

594.

$12 q − ( −6 q ) 12 q − ( −6 q )$

595.

$−9 c − 18 c −9 c − 18 c$

596.

$12x − 4 y − 9 x 12x − 4 y − 9 x$

597.

$3 m 2 + 7 n 2 − 3 m 2 3 m 2 + 7 n 2 − 3 m 2$

598.

$6 x 2 y − 4 x + 8 x y 2 6 x 2 y − 4 x + 8 x y 2$

599.

$13a + b 13a + b$

In the following exercises, add or subtract the polynomials.

600.

$( 5 x 2 + 12 x + 1 ) + ( 6 x 2 − 8 x + 3 ) ( 5 x 2 + 12 x + 1 ) + ( 6 x 2 − 8 x + 3 )$

601.

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

602.

$( 10 m 2 − 8 m − 1 ) − ( 5 m 2 + m − 2 ) ( 10 m 2 − 8 m − 1 ) − ( 5 m 2 + m − 2 )$

603.

$( 7 y 2 − 8 y ) − ( y − 4 ) ( 7 y 2 − 8 y ) − ( y − 4 )$

604.

Subtract
$(3s2+10)from(15s2−2s+8)(3s2+10)from(15s2−2s+8)$

605.

Find the sum of $(a2+6a+9)and(5a3−7)(a2+6a+9)and(5a3−7)$

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

606.

Evaluate $3y2−y+13y2−y+1$ when:

1. $y=5y=5$
2. $y=−1y=−1$
3. $y=0y=0$
607.

Evaluate $10−12x10−12x$ when:

1. $x=3x=3$
2. $x=0x=0$
3. $x=−1x=−1$
608.

Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial $−16t2+200−16t2+200$ gives the height of a stone $tt$ seconds after it is dropped from the cliff. Find the height after $t=3t=3$ seconds.

609.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $−4p2+460p.−4p2+460p.$ Find the revenue received when $p=75p=75$ dollars.

##### Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

610.

$10 4 10 4$

611.

$17 1 17 1$

612.

$( 2 9 ) 2 ( 2 9 ) 2$

613.

$( 0.5 ) 3 ( 0.5 ) 3$

614.

$( −2 ) 6 ( −2 ) 6$

615.

$− 2 6 − 2 6$

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

616.

$x 4 · x 3 x 4 · x 3$

617.

$p 15 · p 16 p 15 · p 16$

618.

$4 10 · 4 6 4 10 · 4 6$

619.

$8 · 8 5 8 · 8 5$

620.

$n · n 2 · n 4 n · n 2 · n 4$

621.

$y c · y 3 y c · y 3$

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

622.

$( m 3 ) 5 ( m 3 ) 5$

623.

$( 5 3 ) 2 ( 5 3 ) 2$

624.

$( y 4 ) x ( y 4 ) x$

625.

$( 3 r ) s ( 3 r ) s$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

626.

$( 4 a ) 2 ( 4 a ) 2$

627.

$( −5 y ) 3 ( −5 y ) 3$

628.

$( 2 m n ) 5 ( 2 m n ) 5$

629.

$( 10 x y z ) 3 ( 10 x y z ) 3$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

630.

$( p 2 ) 5 · ( p 3 ) 6 ( p 2 ) 5 · ( p 3 ) 6$

631.

$( 4 a 3 b 2 ) 3 ( 4 a 3 b 2 ) 3$

632.

$( 5 x ) 2 ( 7 x ) ( 5 x ) 2 ( 7 x )$

633.

$( 2 q 3 ) 4 ( 3 q ) 2 ( 2 q 3 ) 4 ( 3 q ) 2$

634.

$( 1 3 x 2 ) 2 ( 1 2 x ) 3 ( 1 3 x 2 ) 2 ( 1 2 x ) 3$

635.

$( 2 5 m 2 n ) 3 ( 2 5 m 2 n ) 3$

Multiply Monomials

In the following exercises 8, multiply the monomials.

636.

$( −15 x 2 ) ( 6 x 4 ) ( −15 x 2 ) ( 6 x 4 )$

637.

$( −9 n 7 ) ( −16 n ) ( −9 n 7 ) ( −16 n )$

638.

$( 7 p 5 q 3 ) ( 8 p q 9 ) ( 7 p 5 q 3 ) ( 8 p q 9 )$

639.

$( 5 9 a b 2 ) ( 27 a b 3 ) ( 5 9 a b 2 ) ( 27 a b 3 )$

##### Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

640.

$7 ( a + 9 ) 7 ( a + 9 )$

641.

$−4 ( y + 13 ) −4 ( y + 13 )$

642.

$−5 ( r − 2 ) −5 ( r − 2 )$

643.

$p ( p + 3 ) p ( p + 3 )$

644.

$− m ( m + 15 ) − m ( m + 15 )$

645.

$−6 u ( 2 u + 7 ) −6 u ( 2 u + 7 )$

646.

$9 ( b 2 + 6 b + 8 ) 9 ( b 2 + 6 b + 8 )$

647.

$3q2(q2−7q+6)3q2(q2−7q+6)$ 3

648.

$( 5 z − 1 ) z ( 5 z − 1 ) z$

649.

$( b − 4 ) · 11 ( b − 4 ) · 11$

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using: the Distributive Property, the FOIL method, the Vertical Method.

650.

$( x − 4 ) ( x + 10 ) ( x − 4 ) ( x + 10 )$

651.

$( 6 y − 7 ) ( 2 y − 5 ) ( 6 y − 7 ) ( 2 y − 5 )$

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

652.

$( x + 3 ) ( x + 9 ) ( x + 3 ) ( x + 9 )$

653.

$( y − 4 ) ( y − 8 ) ( y − 4 ) ( y − 8 )$

654.

$( p − 7 ) ( p + 4 ) ( p − 7 ) ( p + 4 )$

655.

$( q + 16 ) ( q − 3 ) ( q + 16 ) ( q − 3 )$

656.

$( 5 m − 8 ) ( 12 m + 1 ) ( 5 m − 8 ) ( 12 m + 1 )$

657.

$( u 2 + 6 ) ( u 2 − 5 ) ( u 2 + 6 ) ( u 2 − 5 )$

658.

$( 9 x − y ) ( 6 x − 5 ) ( 9 x − y ) ( 6 x − 5 )$

659.

$( 8 m n + 3 ) ( 2 m n − 1 ) ( 8 m n + 3 ) ( 2 m n − 1 )$

Multiply a Trinomial by a Binomial

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

660.

$( n + 1 ) ( n 2 + 5 n − 2 ) ( n + 1 ) ( n 2 + 5 n − 2 )$

661.

$( 3 x − 4 ) ( 6 x 2 + x − 10 ) ( 3 x − 4 ) ( 6 x 2 + x − 10 )$

In the following exercises, multiply. Use either method.

662.

$( y − 2 ) ( y 2 − 8 y + 9 ) ( y − 2 ) ( y 2 − 8 y + 9 )$

663.

$( 7 m + 1 ) ( m 2 − 10 m − 3 ) ( 7 m + 1 ) ( m 2 − 10 m − 3 )$

##### Special Products

Square a Binomial Using the Binomial Squares Pattern

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

664.

$( c + 11 ) 2 ( c + 11 ) 2$

665.

$( q − 15 ) 2 ( q − 15 ) 2$

666.

$( x + 1 3 ) 2 ( x + 1 3 ) 2$

667.

$( 8 u + 1 ) 2 ( 8 u + 1 ) 2$

668.

$( 3 n 3 − 2 ) 2 ( 3 n 3 − 2 ) 2$

669.

$( 4 a − 3 b ) 2 ( 4 a − 3 b ) 2$

Multiply Conjugates Using the Product of Conjugates Pattern

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

670.

$( s − 7 ) ( s + 7 ) ( s − 7 ) ( s + 7 )$

671.

$( y + 2 5 ) ( y − 2 5 ) ( y + 2 5 ) ( y − 2 5 )$

672.

$( 12 c + 13 ) ( 12 c − 13 ) ( 12 c + 13 ) ( 12 c − 13 )$

673.

$( 6 − r ) ( 6 + r ) ( 6 − r ) ( 6 + r )$

674.

$( u + 3 4 v ) ( u − 3 4 v ) ( u + 3 4 v ) ( u − 3 4 v )$

675.

$( 5 p 4 − 4 q 3 ) ( 5 p 4 + 4 q 3 ) ( 5 p 4 − 4 q 3 ) ( 5 p 4 + 4 q 3 )$

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

676.

$( 3 m + 10 ) 2 ( 3 m + 10 ) 2$

677.

$( 6 a + 11 ) ( 6 a − 11 ) ( 6 a + 11 ) ( 6 a − 11 )$

678.

$( 5 x + y ) ( x − 5 y ) ( 5 x + y ) ( x − 5 y )$

679.

$( c 4 + 9 d ) 2 ( c 4 + 9 d ) 2$

680.

$( p 5 + q 5 ) ( p 5 − q 5 ) ( p 5 + q 5 ) ( p 5 − q 5 )$

681.

$( a 2 + 4 b ) ( 4 a − b 2 ) ( a 2 + 4 b ) ( 4 a − b 2 )$

##### Divide Monomials

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

682.

$u 24 u 6 u 24 u 6$

683.

$10 25 10 5 10 25 10 5$

684.

$3 4 3 6 3 4 3 6$

685.

$v 12 v 48 v 12 v 48$

686.

$x x 5 x x 5$

687.

$5 5 8 5 5 8$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

688.

$75 0 75 0$

689.

$x 0 x 0$

690.

$− 12 0 − 12 0$

691.

$( − 12 0 ) ( − 12 0 )$ $( −12 ) 0 ( −12 ) 0$

692.

$25 x 0 25 x 0$

693.

$( 25 x ) 0 ( 25 x ) 0$

694.

$19 n 0 − 25 m 0 19 n 0 − 25 m 0$

695.

$( 19 n ) 0 − ( 25 m ) 0 ( 19 n ) 0 − ( 25 m ) 0$

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

696.

$( 2 5 ) 3 ( 2 5 ) 3$

697.

$( m 3 ) 4 ( m 3 ) 4$

698.

$( r s ) 8 ( r s ) 8$

699.

$( x 2 y ) 6 ( x 2 y ) 6$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

700.

$( x 3 ) 5 x 9 ( x 3 ) 5 x 9$

701.

$n 10 ( n 5 ) 2 n 10 ( n 5 ) 2$

702.

$( q 6 q 8 ) 3 ( q 6 q 8 ) 3$

703.

$( r 8 r 3 ) 4 ( r 8 r 3 ) 4$

704.

$( c 2 d 5 ) 9 ( c 2 d 5 ) 9$

705.

$( 3 x 4 2 y 2 ) 5 ( 3 x 4 2 y 2 ) 5$

706.

$( v 3 v 9 v 6 ) 4 ( v 3 v 9 v 6 ) 4$

707.

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

Divide Monomials

In the following exercises, divide the monomials.

708.

$−65 y 14 ÷ 5 y 2 −65 y 14 ÷ 5 y 2$

709.

$64 a 5 b 9 −16 a 10 b 3 64 a 5 b 9 −16 a 10 b 3$

710.

$144 x 15 y 8 z 3 18 x 10 y 2 z 12 144 x 15 y 8 z 3 18 x 10 y 2 z 12$

711.

$( 8 p 6 q 2 ) ( 9 p 3 q 5 ) 16 p 8 q 7 ( 8 p 6 q 2 ) ( 9 p 3 q 5 ) 16 p 8 q 7$

##### Divide Polynomials

Divide a Polynomial by a Monomial

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

712.

$42 z 2 − 18 z 6 42 z 2 − 18 z 6$

713.

$( 35 x 2 − 75 x ) ÷ 5 x ( 35 x 2 − 75 x ) ÷ 5 x$

714.

$81 n 4 + 105 n 2 −3 81 n 4 + 105 n 2 −3$

715.

$550 p 6 − 300 p 4 10 p 3 550 p 6 − 300 p 4 10 p 3$

716.

$( 63 x y 3 + 56 x 2 y 4 ) ÷ ( 7 x y ) ( 63 x y 3 + 56 x 2 y 4 ) ÷ ( 7 x y )$

717.

$96 a 5 b 2 − 48 a 4 b 3 − 56 a 2 b 4 8 a b 2 96 a 5 b 2 − 48 a 4 b 3 − 56 a 2 b 4 8 a b 2$

718.

$57 m 2 − 12 m + 1 −3 m 57 m 2 − 12 m + 1 −3 m$

719.

$105 y 5 + 50 y 3 − 5 y 5 y 3 105 y 5 + 50 y 3 − 5 y 5 y 3$

Divide a Polynomial by a Binomial

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

720.

$( k 2 − 2 k − 99 ) ÷ ( k + 9 ) ( k 2 − 2 k − 99 ) ÷ ( k + 9 )$

721.

$( v 2 − 16 v + 64 ) ÷ ( v − 8 ) ( v 2 − 16 v + 64 ) ÷ ( v − 8 )$

722.

$( 3 x 2 − 8 x − 35 ) ÷ ( x − 5 ) ( 3 x 2 − 8 x − 35 ) ÷ ( x − 5 )$

723.

$( n 2 − 3 n − 14 ) ÷ ( n + 3 ) ( n 2 − 3 n − 14 ) ÷ ( n + 3 )$

724.

$( 4 m 3 + m − 5 ) ÷ ( m − 1 ) ( 4 m 3 + m − 5 ) ÷ ( m − 1 )$

725.

$( u 3 − 8 ) ÷ ( u − 2 ) ( u 3 − 8 ) ÷ ( u − 2 )$

##### Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

726.

$9 −2 9 −2$

727.

$( −5 ) −3 ( −5 ) −3$

728.

$3 · 4 −3 3 · 4 −3$

729.

$( 6 u ) −3 ( 6 u ) −3$

730.

$( 2 5 ) −1 ( 2 5 ) −1$

731.

$( 3 4 ) −2 ( 3 4 ) −2$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

732.

$p −2 · p 8 p −2 · p 8$

733.

$q −6 · q −5 q −6 · q −5$

734.

$( c −2 d ) ( c −3 d −2 ) ( c −2 d ) ( c −3 d −2 )$

735.

$( y 8 ) −1 ( y 8 ) −1$

736.

$( q −4 ) −3 ( q −4 ) −3$

737.

$a 8 a 12 a 8 a 12$

738.

$n 5 n −4 n 5 n −4$

739.

$r −2 r −3 r −2 r −3$

Convert from Decimal Notation to Scientific Notation

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

740.

8,500,000

741.

0.00429

742.

The thickness of a dime is about 0.053 inches.

743.

In 2015, the population of the world was about 7,200,000,000 people.

Convert Scientific Notation to Decimal Form

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

744.

$3.8 × 10 5 3.8 × 10 5$

745.

$1.5 × 10 10 1.5 × 10 10$

746.

$9.1 × 10 −7 9.1 × 10 −7$

747.

$5.5 × 10 −1 5.5 × 10 −1$

Multiply and Divide Using Scientific Notation

In the following exercises, multiply and write your answer in decimal form.

748.

$( 2 × 10 5 ) ( 4 × 10 −3 ) ( 2 × 10 5 ) ( 4 × 10 −3 )$

749.

$( 3.5 × 10 −2 ) ( 6.2 × 10 −1 ) ( 3.5 × 10 −2 ) ( 6.2 × 10 −1 )$

In the following exercises, divide and write your answer in decimal form.

750.

$8 × 10 5 4 × 10 −1 8 × 10 5 4 × 10 −1$

751.

$9 × 10 −5 3 × 10 2 9 × 10 −5 3 × 10 2$

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