Intermediate Algebra

# Review Exercises

Intermediate AlgebraReview Exercises

### Review Exercises

##### Add and Subtract Polynomials

Determine the Degree of Polynomials

In the following exercises, determine the type of polynomial.

342.

$16x2−40x−2516x2−40x−25$

343.

$5m+95m+9$

344.

$−15−15$

345.

$y2+6y3+9y4y2+6y3+9y4$

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

346.

$4p+11p4p+11p$

347.

$−8y3−5y3−8y3−5y3$

348.

$(4a2+9a−11)+(6a2−5a+10)(4a2+9a−11)+(6a2−5a+10)$

349.

$(8m2+12m−5)−(2m2−7m−1)(8m2+12m−5)−(2m2−7m−1)$

350.

$(y2−3y+12)+(5y2−9)(y2−3y+12)+(5y2−9)$

351.

$(5u2+8u)−(4u−7)(5u2+8u)−(4u−7)$

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.

$17mn2−(−9mn2)+3mn217mn2−(−9mn2)+3mn2$

355.

$18a−7b−21a18a−7b−21a$

356.

$2pq2−5p−3q22pq2−5p−3q2$

357.

$(6a2+7)+(2a2−5a−9)(6a2+7)+(2a2−5a−9)$

358.

$(3p2−4p−9)+(5p2+14)(3p2−4p−9)+(5p2+14)$

359.

$(7m2−2m−5)−(4m2+m−8)(7m2−2m−5)−(4m2+m−8)$

360.

$(7b2−4b+3)−(8b2−5b−7)(7b2−4b+3)−(8b2−5b−7)$

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.

$(x3−x2y)−(4xy2−y3)+(3x2y−xy2)(x3−x2y)−(4xy2−y3)+(3x2y−xy2)$

364.

$(x3−2x2y)−(xy2−3y3)−(x2y−4xy2)(x3−2x2y)−(xy2−3y3)−(x2y−4xy2)$

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.

Add and Subtract Polynomial Functions

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.

$p3·p10p3·p10$

372.

$2·262·26$

373.

$a·a2·a3a·a2·a3$

374.

$x·x8x·x8$

375.

$ya·ybya·yb$

376.

$28222822$

377.

$a6aa6a$

378.

$n3n12n3n12$

379.

$1x51x5$

380.

$3030$

381.

$y0y0$

382.

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

383.

$12a0−15b012a0−15b0$

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

384.

$6−26−2$

385.

$(−10)−3(−10)−3$

386.

$5·2−45·2−4$

387.

$(8n)−1(8n)−1$

388.

$y−5y−5$

389.

$10−310−3$

390.

$1a−41a−4$

391.

$16−216−2$

392.

$−5−3−5−3$

393.

$(−15)−3(−15)−3$

394.

$−(12)−3−(12)−3$

395.

$(−5)−3(−5)−3$

396.

$(59)−2(59)−2$

397.

$(−3x)−3(−3x)−3$

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

398.

$(y4)3(y4)3$

399.

$(32)5(32)5$

400.

$(a10)y(a10)y$

401.

$x−3·x9x−3·x9$

402.

$r−5·r−4r−5·r−4$

403.

$(uv−3)(u−4v−2)(uv−3)(u−4v−2)$

404.

$(m5)−1(m5)−1$

405.

$p5·p−2·p−4p5·p−2·p−4$

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

406.

$(k−2)−3(k−2)−3$

407.

$q4q20q4q20$

408.

$b8b−2b8b−2$

409.

$n−3n−5n−3n−5$

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

410.

$(−5ab)3(−5ab)3$

411.

$(−4pq)0(−4pq)0$

412.

$(−6x3)−2(−6x3)−2$

413.

$(3y−4)2(3y−4)2$

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

414.

$(35x)−2(35x)−2$

415.

$(3xy2z)4(3xy2z)4$

416.

$(4p−3q2)2(4p−3q2)2$

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

417.

$(x2y)2(3xy5)3(x2y)2(3xy5)3$

418.

$(−3a−2)4(2a4)2(−6a2)3(−3a−2)4(2a4)2(−6a2)3$

419.

$(3xy34x4y−2)2(6xy48x3y−2)−1(3xy34x4y−2)2(6xy48x3y−2)−1$

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

420.

$2.5682.568$

421.

5,300,000

422.

$0.008140.00814$

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

423.

$2.9×1042.9×104$

424.

$3.75×10−13.75×10−1$

425.

$9.413×10−59.413×10−5$

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

426.

$(3×107)(2×10−4)(3×107)(2×10−4)$

427.

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

428.

$6×1092×10−16×1092×10−1$

429.

$9×10−31×10−69×10−31×10−6$

##### Multiply Polynomials

Multiply Monomials

In the following exercises, multiply the monomials.

430.

$(−6p4)(9p)(−6p4)(9p)$

431.

$(13c2)(30c8)(13c2)(30c8)$

432.

$(8x2y5)(7xy6)(8x2y5)(7xy6)$

433.

$(23m3n6)(16m4n4)(23m3n6)(16m4n4)$

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

434.

$7(10−x)7(10−x)$

435.

$a2(a2−9a−36)a2(a2−9a−36)$

436.

$−5y(125y3−1)−5y(125y3−1)$

437.

$(4n−5)(2n3)(4n−5)(2n3)$

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.

$(3x+1)(2x−7)(3x+1)(2x−7)$

441.

$(6p−11)(3p−10)(6p−11)(3p−10)$

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.

$(5u−3)(u+8)(5u−3)(u+8)$

445.

$(2y−9)(5y−7)(2y−9)(5y−7)$

446.

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

447.

$(x−8)(x+9)(x−8)(x+9)$

448.

$(3c+1)(9c−4)(3c+1)(9c−4)$

449.

$(10a−1)(3a−3)(10a−1)(3a−3)$

Multiply a Polynomial by a Polynomial

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

450.

$(x+1)(x2−3x−21)(x+1)(x2−3x−21)$

451.

$(5b−2)(3b2+b−9)(5b−2)(3b2+b−9)$

In the following exercises, multiply. Use either method.

452.

$(m+6)(m2−7m−30)(m+6)(m2−7m−30)$

453.

$(4y−1)(6y2−12y+5)(4y−1)(6y2−12y+5)$

Multiply Special Products

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

454.

$(2x−y)2(2x−y)2$

455.

$(x+34)2(x+34)2$

456.

$(8p3−3)2(8p3−3)2$

457.

$(5p+7q)2(5p+7q)2$

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

458.

$(3y+5)(3y−5)(3y+5)(3y−5)$

459.

$(6x+y)(6x−y)(6x+y)(6x−y)$

460.

$(a+23b)(a−23b)(a+23b)(a−23b)$

461.

$(12x3−7y2)(12x3+7y2)(12x3−7y2)(12x3+7y2)$

462.

$(13a2−8b4)(13a2+8b4)(13a2−8b4)(13a2+8b4)$

##### Divide Monomials

Divide Monomials

In the following exercises, divide the monomials.

463.

$72p12÷8p372p12÷8p3$

464.

$−26a8÷(2a2)−26a8÷(2a2)$

465.

$45y6−15y1045y6−15y10$

466.

$−30x8−36x9−30x8−36x9$

467.

$28a9b7a4b328a9b7a4b3$

468.

$11u6v355u2v811u6v355u2v8$

469.

$(5m9n3)(8m3n2)(10mn4)(m2n5)(5m9n3)(8m3n2)(10mn4)(m2n5)$

470.

$(42r2s4)(54rs2)(6rs3)(9s)(42r2s4)(54rs2)(6rs3)(9s)$

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial

471.

$(54y4−24y3)÷(−6y2)(54y4−24y3)÷(−6y2)$

472.

$63x3y2−99x2y3−45x4y39x2y263x3y2−99x2y3−45x4y39x2y2$

473.

$12x2+4x−3−4x12x2+4x−3−4x$

Divide Polynomials using Long Division

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

474.

$(4x2−21x−18)÷(x−6)(4x2−21x−18)÷(x−6)$

475.

$(y2+2y+18)÷(y+5)(y2+2y+18)÷(y+5)$

476.

$(n3−2n2−6n+27)÷(n+3)(n3−2n2−6n+27)÷(n+3)$

477.

$(a3−1)÷(a+1)(a3−1)÷(a+1)$

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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