5.7.1: Review Exercises

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Review Exercises

Add and Subtract Polynomials

Types of Polynomials

In the following exercises, determine the type of polynomial.

342.

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

343.

$5 m + 9$

344.

$−15$

345.

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

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

346.

$4 p + 11 p$

347.

$−8 y^{3} - 5 y^{3}$

348.

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

349.

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

350.

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

351.

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

352.

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

353.

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

In the following exercises, simplify.

354.

$17 m n^{2} - (−9 m n^{2}) + 3 m n^{2}$

355.

$18 a - 7 b - 21 a$

356.

$2 p q^{2} - 5 p - 3 q^{2}$

357.

$(6 a^{2} + 7) + (2 a^{2} - 5 a - 9)$

358.

$(3 p^{2} - 4 p - 9) + (5 p^{2} + 14)$

359.

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

360.

$(7 b^{2} - 4 b + 3) - (8 b^{2} - 5 b - 7)$

361.

Subtract $(8 y^{2} - y + 9)$ from $(11 y^{2} - 9 y - 5)$

362.

Find the difference of $(z^{2} - 4 z - 12)$ and $(3 z^{2} + 2 z - 11)$

363.

$(x^{3} - x^{2} y) - (4 x y^{2} - y^{3}) + (3 x^{2} y - x y^{2})$

364.

$(x^{3} - 2 x^{2} y) - (x y^{2} - 3 y^{3}) - (x^{2} y - 4 x y^{2})$

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 $f (x) = 7 x^{2} - 3 x + 5$ find:
ⓐ $f (5)$ ⓑ $f (−2)$ ⓒ $f (0)$

366.

For the function $g (x) = 15 - 16 x^{2},$ find:
ⓐ $g (−1)$ ⓑ $g (0)$ ⓒ $g (2)$

367.

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

368.

A manufacturer of the latest soccer shoes has found that the revenue received from selling the shoes at a cost of $p$ dollars each is given by the polynomial $R (p) = −5 p^{2} + 360 p .$ Find the revenue received when $p = 10$ dollars.

Add and Subtract Polynomial Functions

In the following exercises, find ⓐ (f + g)(x) ⓑ (f + g)(3) ⓒ (f − g)(x) ⓓ (f − g)(−2)

369.

$f (x) = 2 x^{2} - 4 x - 7$ and $g (x) = 2 x^{2} - x + 5$

370.

$f (x) = 4 x^{3} - 3 x^{2} + x - 1$ and $g (x) = 8 x^{3} - 1$

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.

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

372.

$2 \cdot 2^{6}$

373.

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

374.

$x \cdot x^{8}$

375.

$y^{a} \cdot y^{b}$

376.

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

377.

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

378.

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

379.

$\frac{1}{x^{5}}$

380.

$3^{0}$

381.

$y^{0}$

382.

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

383.

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

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

384.

$6^{−2}$

385.

${(−10)}^{−3}$

386.

$5 \cdot 2^{−4}$

387.

${(8 n)}^{−1}$

388.

$y^{−5}$

389.

$10^{−3}$

390.

$\frac{1}{a^{−4}}$

391.

$\frac{1}{6^{−2}}$

392.

$− 5^{−3}$

393.

${(- \frac{1}{5})}^{−3}$

394.

$− {(\frac{1}{2})}^{−3}$

395.

${(−5)}^{−3}$

396.

${(\frac{5}{9})}^{−2}$

397.

${(- \frac{3}{x})}^{−3}$

In the following exercises, simplify each expression using the Product Property.

398.

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

399.

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

400.

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

401.

$x^{−3} \cdot x^{9}$

402.

$r^{−5} \cdot r^{−4}$

403.

$(u v^{−3}) (u^{−4} v^{−2})$

404.

${(m^{5})}^{−1}$

405.

$p^{5} \cdot p^{−2} \cdot p^{−4}$

In the following exercises, simplify each expression using the Power Property.

406.

${(k^{−2})}^{−3}$

407.

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

408.

$\frac{b^{8}}{b^{−2}}$

409.

$\frac{n^{−3}}{n^{−5}}$

In the following exercises, simplify each expression using the Product to a Power Property.

410.

${(−5 a b)}^{3}$

411.

${(−4 p q)}^{0}$

412.

${(−6 x^{3})}^{−2}$

413.

${(3 y^{−4})}^{2}$

In the following exercises, simplify each expression using the Quotient to a Power Property.

414.

${(\frac{3}{5 x})}^{−2}$

415.

${(\frac{3 x y^{2}}{z})}^{4}$

416.

${(\frac{4 p^{−3}}{q^{2}})}^{2}$

In the following exercises, simplify each expression by applying several properties.

417.

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

418.

$\frac{{(−3 a^{−2})}^{4} {(2 a^{4})}^{2}}{{(−6 a^{2})}^{3}}$

419.

${(\frac{3 x y^{3}}{4 x^{4} y^{−2}})}^{2} {(\frac{6 x y^{4}}{8 x^{3} y^{−2}})}^{−1}$

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

420.

$2.568$

421.

5,300,000

422.

$0.00814$

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

423.

$2.9 \times 10^{4}$

424.

$3.75 \times 10^{−1}$

425.

$9.413 \times 10^{−5}$

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

426.

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

427.

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

428.

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

429.

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

Multiply Polynomials

Multiply Monomials

In the following exercises, multiply the monomials.

430.

$(−6 p^{4}) (9 p)$

431.

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

432.

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

433.

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

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

434.

$7 (10 - x)$

435.

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

436.

$−5 y (125 y^{3} - 1)$

437.

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

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using:

ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

438.

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

439.

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

440.

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

441.

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

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

442.

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

443.

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

444.

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

445.

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

446.

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

447.

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

448.

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

449.

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

Multiply a Polynomial by a Polynomial

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

450.

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

451.

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

In the following exercises, multiply. Use either method.

452.

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

453.

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

Multiply Special Products

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

454.

${(2 x - y)}^{2}$

455.

${(x + \frac{3}{4})}^{2}$

456.

${(8 p^{3} - 3)}^{2}$

457.

${(5 p + 7 q)}^{2}$

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

458.

$(3 y + 5) (3 y - 5)$

459.

$(6 x + y) (6 x - y)$

460.

$(a + \frac{2}{3} b) (a - \frac{2}{3} b)$

461.

$(12 x^{3} - 7 y^{2}) (12 x^{3} + 7 y^{2})$

462.

$(13 a^{2} - 8 b^{4}) (13 a^{2} + 8 b^{4})$

Divide Monomials

In the following exercises, divide the monomials.

463.

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

464.

$−26 a^{8} \div (2 a^{2})$

465.

$\frac{45 y^{6}}{−15 y^{10}}$

466.

$\frac{−30 x^{8}}{−36 x^{9}}$

467.

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

468.

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

469.

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

470.

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

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial

471.

$(54 y^{4} - 24 y^{3}) \div (−6 y^{2})$

472.

$\frac{63 x^{3} y^{2} - 99 x^{2} y^{3} - 45 x^{4} y^{3}}{9 x^{2} y^{2}}$

473.

$\frac{12 x^{2} + 4 x - 3}{−4 x}$

Divide Polynomials using Long Division

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

474.

$(4 x^{2} - 21 x - 18) \div (x - 6)$

475.

$(y^{2} + 2 y + 18) \div (y + 5)$

476.

$(n^{3} - 2 n^{2} - 6 n + 27) \div (n + 3)$

477.

$(a^{3} - 1) \div (a + 1)$

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

478.

$x^{3} - 3 x^{2} - 4 x + 12$ is divided by $x + 2$

479.

$2 x^{3} - 11 x^{2} + 11 x + 12$ is divided by $x - 3$

480.

$x^{4} + x^{2} + 6 x - 10$ is divided by $x + 2$

Divide Polynomial Functions

In the following exercises, divide.

481.

For functions $f (x) = x^{2} - 15 x + 54$ and $g (x) = x - 9,$ find ⓐ $(\frac{f}{g}) (x)$
ⓑ $(\frac{f}{g}) (−2)$

482.

For functions $f (x) = x^{3} + x^{2} - 7 x + 2$ and $g (x) = x - 2,$ find ⓐ $(\frac{f}{g}) (x)$
ⓑ $(\frac{f}{g}) (3)$

Use the Remainder and Factor Theorem

In the following exercises, use the Remainder Theorem to find the remainder.

483.

$f (x) = x^{3} - 4 x - 9$ is divided by $x + 2$

484.

$f (x) = 2 x^{3} - 6 x - 24$ divided by $x - 3$

In the following exercises, use the Factor Theorem to determine if $x - c$ is a factor of the polynomial function.

485.

Determine whether $x - 2$ is a factor of $x^{3} - 7 x^{2} + 7 x - 6$ .

486.

Determine whether $x - 3$ is a factor of $x^{3} - 7 x^{2} + 11 x + 3$ .

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