Prealgebra

Review Exercises

PrealgebraReview 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.

494.

$y 2 + 8 y − 20 y 2 + 8 y − 20$

495.

$−6 a 4 −6 a 4$

496.

$9 x 3 − 1 9 x 3 − 1$

497.

$n 3 − 3 n 2 + 3 n − 1 n 3 − 3 n 2 + 3 n − 1$

Determine the Degree of Polynomials

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

498.

$16 x 2 − 40 x − 25 16 x 2 − 40 x − 25$

499.

$5 m + 9 5 m + 9$

500.

$−15 −15$

501.

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

In the following exercises, add or subtract the monomials.

502.

$4 p + 11 p 4 p + 11 p$

503.

$−8 y 3 − 5 y 3 −8 y 3 − 5 y 3$

504.

Add $4n5,−n5,−6n54n5,−n5,−6n5$

505.

Subtract $10x210x2$ from $3x23x2$

In the following exercises, add or subtract the polynomials.

506.

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

507.

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

508.

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

509.

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

510.

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

511.

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

Evaluate a Polynomial for a Given Value of the Variable

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

512.

$200x−15x2200x−15x2$ when $x=5x=5$

513.

$200x−15x2200x−15x2$ when $x=0x=0$

514.

$200x−15x2200x−15x2$ when $x=15x=15$

515.

$5+40x−12x25+40x−12x2$ when $x=10x=10$

516.

$5+40x−12x25+40x−12x2$ when $x=−4x=−4$

517.

$5+40x−12x25+40x−12x2$ when $x=0x=0$

518.

A pair of glasses is dropped off a bridge $640640$ feet above a river. The polynomial $−16t2+640−16t2+640$ gives the height of the glasses $tt$ seconds after they were dropped. Find the height of the glasses when $t=6.t=6.$

519.

The fuel efficiency (in miles per gallon) of a bus going at a speed of $xx$ miles per hour is given by the polynomial $−1160x2+12x.−1160x2+12x.$ Find the fuel efficiency when $x=20x=20$ mph.

Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

520.

$6 3 6 3$

521.

$( 1 2 ) 4 ( 1 2 ) 4$

522.

$( −0.5 ) 2 ( −0.5 ) 2$

523.

$− 3 2 − 3 2$

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression.

524.

$p 3 · p 10 p 3 · p 10$

525.

$2 · 2 6 2 · 2 6$

526.

$a · a 2 · a 3 a · a 2 · a 3$

527.

$x · x 8 x · x 8$

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression.

528.

$( y 4 ) 3 ( y 4 ) 3$

529.

$( r 3 ) 2 ( r 3 ) 2$

530.

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

531.

$( a 10 ) y ( a 10 ) y$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

532.

$( 8 n ) 2 ( 8 n ) 2$

533.

$( −5 x ) 3 ( −5 x ) 3$

534.

$( 2 a b ) 8 ( 2 a b ) 8$

535.

$( −10 m n p ) 4 ( −10 m n p ) 4$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

536.

$( 3 a 5 ) 3 ( 3 a 5 ) 3$

537.

$( 4 y ) 2 ( 8 y ) ( 4 y ) 2 ( 8 y )$

538.

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

539.

$( 5 s t 2 ) 3 ( 2 s 3 t 4 ) 2 ( 5 s t 2 ) 3 ( 2 s 3 t 4 ) 2$

Multiply Monomials

In the following exercises, multiply the monomials.

540.

$( −6 p 4 ) ( 9 p ) ( −6 p 4 ) ( 9 p )$

541.

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

542.

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

543.

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

Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

544.

$7 ( 10 − x ) 7 ( 10 − x )$

545.

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

546.

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

547.

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

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using various methods.

548.

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

549.

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

550.

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

551.

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

552.

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

553.

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

554.

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

555.

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

556.

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

557.

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

558.

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

559.

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

Multiply a Trinomial by a Binomial

In the following exercises, multiply using any method.

560.

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

561.

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

562.

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

563.

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

Divide Monomials

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

564.

$2 8 2 2 2 8 2 2$

565.

$a 6 a a 6 a$

566.

$n 3 n 12 n 3 n 12$

567.

$x x 5 x x 5$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

568.

$3 0 3 0$

569.

$y 0 y 0$

570.

$( 14 t ) 0 ( 14 t ) 0$

571.

$12 a 0 − 15 b 0 12 a 0 − 15 b 0$

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

572.

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

573.

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

574.

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

575.

$( s 10 t ) 2 ( s 10 t ) 2$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

576.

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

577.

$u 3 u 2 · u 4 u 3 u 2 · u 4$

578.

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

579.

$( p 4 · p 5 p 3 ) 2 ( p 4 · p 5 p 3 ) 2$

580.

$( n 5 ) 3 ( n 2 ) 8 ( n 5 ) 3 ( n 2 ) 8$

581.

$( 5 s 2 4 t ) 3 ( 5 s 2 4 t ) 3$

Divide Monomials

In the following exercises, divide the monomials.

582.

$72 p 12 ÷ 8 p 3 72 p 12 ÷ 8 p 3$

583.

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

584.

$45 y 6 −15 y 10 45 y 6 −15 y 10$

585.

$−30 x 8 −36 x 9 −30 x 8 −36 x 9$

586.

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

587.

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

588.

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

589.

$42 r 2 s 4 6 r s 3 − 54 r s 2 9 s 42 r 2 s 4 6 r s 3 − 54 r s 2 9 s$

Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

590.

$6 −2 6 −2$

591.

$( −10 ) −3 ( −10 ) −3$

592.

$5 · 2 −4 5 · 2 −4$

593.

$( 8 n ) −1 ( 8 n ) −1$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

594.

$x −3 · x 9 x −3 · x 9$

595.

$r −5 · r −4 r −5 · r −4$

596.

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

597.

$( m 5 ) −1 ( m 5 ) −1$

598.

$( k −2 ) −3 ( k −2 ) −3$

599.

$q 4 q 20 q 4 q 20$

600.

$b 8 b −2 b 8 b −2$

601.

$n −3 n −5 n −3 n −5$

Convert from Decimal Notation to Scientific Notation

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

602.

$5,300,000 5,300,000$

603.

$0.00814 0.00814$

604.

The thickness of a piece of paper is about $0.0970.097$ millimeter.

605.

According to www.cleanair.com, U.S. businesses use about $21,000,00021,000,000$ tons of paper per year.

Convert Scientific Notation to Decimal Form

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

606.

$2.9 × 10 4 2.9 × 10 4$

607.

$1.5 × 10 8 1.5 × 10 8$

608.

$3.75 × 10 −1 3.75 × 10 −1$

609.

$9.413 × 10 −5 9.413 × 10 −5$

Multiply and Divide Using Scientific Notation

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

610.

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

611.

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

612.

$6 × 10 9 2 × 10 −1 6 × 10 9 2 × 10 −1$

613.

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

Introduction to Factoring Polynomials

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

614.

$5 n , 45 5 n , 45$

615.

$8 a , 72 8 a , 72$

616.

$12 x 2 , 20 x 3 , 36 x 4 12 x 2 , 20 x 3 , 36 x 4$

617.

$9 y 4 , 21 y 5 , 15 y 6 9 y 4 , 21 y 5 , 15 y 6$

Factor the Greatest Common Factor from a Polynomial

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

618.

$16 u − 24 16 u − 24$

619.

$15 r + 35 15 r + 35$

620.

$6 p 2 + 6 p 6 p 2 + 6 p$

621.

$10 c 2 − 10 c 10 c 2 − 10 c$

622.

$−9 a 5 − 9 a 3 −9 a 5 − 9 a 3$

623.

$−7 x 8 − 28 x 3 −7 x 8 − 28 x 3$

624.

$5 y 2 − 55 y + 45 5 y 2 − 55 y + 45$

625.

$2 q 5 − 16 q 3 + 30 q 2 2 q 5 − 16 q 3 + 30 q 2$

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