10.9.1: Review Exercises

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Review Exercises

Add and Subtract Polynomials

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.

$y^{2} + 8 y - 20$

495.

$−6 a^{4}$

496.

$9 x^{3} - 1$

497.

$n^{3} - 3 n^{2} + 3 n - 1$

Determine the Degree of Polynomials

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

498.

$16 x^{2} - 40 x - 25$

499.

$5 m + 9$

500.

$−15$

501.

$y^{2} + 6 y^{3} + 9 y^{4}$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

502.

$4 p + 11 p$

503.

$−8 y^{3} - 5 y^{3}$

504.

Add $4 n^{5}, − n^{5}, − 6 n^{5}$

505.

Subtract $10 x^{2}$ from $3 x^{2}$

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

506.

$(4 a^{2} + 9 a - 11) + (6 a^{2} - 5 a + 10)$

507.

$(8 m^{2} + 12 m - 5) - (2 m^{2} - 7 m - 1)$

508.

$(y^{2} - 3 y + 12) + (5 y^{2} - 9)$

509.

$(5 u^{2} + 8 u) - (4 u - 7)$

510.

Find the sum of $8 q^{3} - 27$ and $q^{2} + 6 q - 2$

511.

Find the difference of $x^{2} + 6 x + 8$ and $x^{2} - 8 x + 15$

Evaluate a Polynomial for a Given Value of the Variable

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

512.

$200 x - \frac{1}{5} x^{2}$ when $x = 5$

513.

$200 x - \frac{1}{5} x^{2}$ when $x = 0$

514.

$200 x - \frac{1}{5} x^{2}$ when $x = 15$

515.

$5 + 40 x - \frac{1}{2} x^{2}$ when $x = 10$

516.

$5 + 40 x - \frac{1}{2} x^{2}$ when $x = −4$

517.

$5 + 40 x - \frac{1}{2} x^{2}$ when $x = 0$

518.

A pair of glasses is dropped off a bridge $640$ feet above a river. The polynomial $−16 t^{2} + 640$ gives the height of the glasses $t$ seconds after they were dropped. Find the height of the glasses when $t = 6 .$

519.

The fuel efficiency (in miles per gallon) of a bus going at a speed of $x$ miles per hour is given by the polynomial $- \frac{1}{160} x^{2} + \frac{1}{2} x .$ Find the fuel efficiency when $x = 20$ mph.

Use Multiplication Properties of Exponents

Simplify Expressions with Exponents

In the following exercises, simplify.

520.

$6^{3}$

521.

${(\frac{1}{2})}^{4}$

522.

${(−0.5)}^{2}$

523.

$- 3^{2}$

Simplify Expressions Using the Product Property of Exponents

In the following exercises, simplify each expression.

524.

$p^{3} \cdot p^{10}$

525.

$2 \cdot 2^{6}$

526.

$a \cdot a^{2} \cdot a^{3}$

527.

$x \cdot x^{8}$

Simplify Expressions Using the Power Property of Exponents

In the following exercises, simplify each expression.

528.

${(y^{4})}^{3}$

529.

${(r^{3})}^{2}$

530.

${(3^{2})}^{5}$

531.

${(a^{10})}^{y}$

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

532.

${(8 n)}^{2}$

533.

${(−5 x)}^{3}$

534.

${(2 a b)}^{8}$

535.

${(−10 m n p)}^{4}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

536.

${(3 a^{5})}^{3}$

537.

${(4 y)}^{2} (8 y)$

538.

${(x^{3})}^{5} {(x^{2})}^{3}$

539.

${(5 s t^{2})}^{3} {(2 s^{3} t^{4})}^{2}$

Multiply Monomials

In the following exercises, multiply the monomials.

540.

$(−6 p^{4}) (9 p)$

541.

$(\frac{1}{3} c^{2}) (30 c^{8})$

542.

$(8 x^{2} y^{5}) (7 x y^{6})$

543.

$(\frac{2}{3} m^{3} n^{6}) (\frac{1}{6} m^{4} n^{4})$

Multiply Polynomials

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

544.

$7 (10 - x)$

545.

$a^{2} (a^{2} - 9 a - 36)$

546.

$−5 y (125 y^{3} - 1)$

547.

$(4 n - 5) (2 n^{3})$

Multiply a Binomial by a Binomial

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

548.

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

549.

$(y - 4) (y + 12)$

550.

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

551.

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

552.

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

553.

$(k + 6) (k - 9)$

554.

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

555.

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

556.

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

557.

$(x - 8) (x + 9)$

558.

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

559.

$(10 a - 1) (3 a - 3)$

Multiply a Trinomial by a Binomial

In the following exercises, multiply using any method.

560.

$(x + 1) (x^{2} - 3 x - 21)$

561.

$(5 b - 2) (3 b^{2} + b - 9)$

562.

$(m + 6) (m^{2} - 7 m - 30)$

563.

$(4 y - 1) (6 y^{2} - 12 y + 5)$

Divide Monomials

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

564.

$\frac{2^{8}}{2^{2}}$

565.

$\frac{a^{6}}{a}$

566.

$\frac{n^{3}}{n^{12}}$

567.

$\frac{x}{x^{5}}$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

568.

$3^{0}$

569.

$y^{0}$

570.

${(14 t)}^{0}$

571.

$12 a^{0} - 15 b^{0}$

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

572.

${(\frac{3}{5})}^{2}$

573.

${(\frac{x}{2})}^{5}$

574.

${(\frac{5 m}{n})}^{3}$

575.

${(\frac{s}{10 t})}^{2}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

576.

$\frac{{(a^{3})}^{2}}{a^{4}}$

577.

$\frac{u^{3}}{u^{2} \cdot u^{4}}$

578.

${(\frac{x}{x^{9}})}^{5}$

579.

${(\frac{p^{4} \cdot p^{5}}{p^{3}})}^{2}$

580.

$\frac{{(n^{5})}^{3}}{{(n^{2})}^{8}}$

581.

${(\frac{5 s^{2}}{4 t})}^{3}$

Divide Monomials

In the following exercises, divide the monomials.

582.

$72 p^{12} \div 8 p^{3}$

583.

$−26 a^{8} \div (2 a^{2})$

584.

$\frac{45 y^{6}}{−15 y^{10}}$

585.

$\frac{−30 x^{8}}{−36 x^{9}}$

586.

$\frac{28 a^{9} b}{7 a^{4} b^{3}}$

587.

$\frac{11 u^{6} v^{3}}{55 u^{2} v^{8}}$

588.

$\frac{(5 m^{9} n^{3}) (8 m^{3} n^{2})}{(10 m n^{4}) (m^{2} n^{5})}$

589.

$\frac{42 r^{2} s^{4}}{6 r s^{3}} - \frac{54 r s^{2}}{9 s}$

Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent

In the following exercises, simplify.

590.

$6^{−2}$

591.

${(−10)}^{−3}$

592.

$5 \cdot 2^{−4}$

593.

${(8 n)}^{−1}$

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

594.

$x^{−3} \cdot x^{9}$

595.

$r^{−5} \cdot r^{−4}$

596.

$(u v^{−3}) (u^{−4} v^{−2})$

597.

${(m^{5})}^{−1}$

598.

${(k^{−2})}^{−3}$

599.

$\frac{q^{4}}{q^{20}}$

600.

$\frac{b^{8}}{b^{−2}}$

601.

$\frac{n^{−3}}{n^{−5}}$

Convert from Decimal Notation to Scientific Notation

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

602.

$5,300,000$

603.

$0.00814$

604.

The thickness of a piece of paper is about $0.097$ millimeter.

605.

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

Convert Scientific Notation to Decimal Form

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

606.

$2.9 \times 10^{4}$

607.

$1.5 \times 10^{8}$

608.

$3.75 \times 10^{−1}$

609.

$9.413 \times 10^{−5}$

Multiply and Divide Using Scientific Notation

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

610.

$(3 \times 10^{7}) (2 \times 10^{−4})$

611.

$(1.5 \times 10^{−3}) (4.8 \times 10^{−1})$

612.

$\frac{6 \times 10^{9}}{2 \times 10^{−1}}$

613.

$\frac{9 \times 10^{−3}}{1 \times 10^{−6}}$

Introduction to Factoring Polynomials

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

614.

$5 n, 45$

615.

$8 a, 72$

616.

$12 x^{2}, 20 x^{3}, 36 x^{4}$

617.

$9 y^{4}, 21 y^{5}, 15 y^{6}$

Factor the Greatest Common Factor from a Polynomial

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

618.

$16 u - 24$

619.

$15 r + 35$

620.

$6 p^{2} + 6 p$

621.

$10 c^{2} - 10 c$

622.

$−9 a^{5} - 9 a^{3}$

623.

$−7 x^{8} - 28 x^{3}$

624.

$5 y^{2} - 55 y + 45$

625.

$2 q^{5} - 16 q^{3} + 30 q^{2}$

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Review Exercises

Add and Subtract Polynomials