Chapter 7

Multiplying by One or More Terms

Multiplying algebraic expressions is much like multiplying numbers, but the introduction of variables makes the process just a bit more interesting. Products involving variables call on the rules of exponents. And, because of the commutative property of addition and multiplication, arrangements and rearrangements of terms and factors can make the process simpler.

The Problems You'll Work On

When multiplying by one or more terms, you deal with the following in this chapter:

  • Distributing terms with one or more factors over two or more terms — multiplication over sums and differences
  • Distributing division over sums and differences and dividing each term in the parentheses
  • Distributing binomials over binomials or trinomials and then combining like terms
  • Multiplying binomials using FOIL: First, Outer, Inner, Last
  • Using Pascal's triangle to find powers of binomials
  • Finding products of binomials times trinomials that create sums and differences of cubes

What to Watch Out For

With all the distributing and multiplying, don't overlook the following:

  • Applying the rules of exponents to all terms when distributing variables over several terms
  • Changing the sign of each term when distributing a negative factor over several terms
  • Combining the outer and inner terms correctly when applying FOIL
  • Starting with the zero power when assigning powers of the second term to the pattern in Pascal's triangle

Distributing One Term Over Sums and Differences

311–315 Distribute the number over the terms in the parentheses.

311. 3(2x + 4)

312. −4(5y − 6)

313. 7(x2 − 2x + 3)

314. image

315. image

Distributing Using Division

316–320 Perform the division by dividing each term in the numerator by the term in the denominator.

316. image

317. image

318. image

319. image

320. image

Multiplying Binomials Using Distributing

321–325 Distribute the first binomial over the second binomial and simplify.

321. (a + 1)(x − 2)

322. (y − 4)(z2 + 7)

323. (x + 2)(z − 2)

324. (x2 − 7)(x3 − 8)

325. (x2 + y4)(x2 − y4)

Multiplying Binomials Using FOIL

326–335 Multiply the binomials using “FOIL.”

326. (x − 3)(x + 2)

327. (y + 6)(y + 4)

328. (2x − 3)(3x − 2)

329. (z − 4)(3z − 8)

330. (5x + 3)(4x − 2)

331. (3y − 4)(7y + 4)

332. (x2 − 1)(x2 + 1)

333. (2y3 + 1)(3y3 − 2)

334. (8x − 7)(8x + 7)

335. (2z2 + 3)(2z2 − 3)

Distributing Binomials Over Trinomials

336–340 Distribute the binomial over the trinomial and simplify.

336. (x + 3)(x2 − 2x + 1)

337. (y − 2)(y2 + 3y + 4)

338. (2z + 1)(z2 + z + 7)

339. (4x − 3)(2x2 + 2x + 1)

340. (y + 7)(3y2 − 7y + 5)

Squaring Binomials

341–345 Square the binomials.

341. (x + 5)2

342. (y − 6)2

343. (4z + 3)2

344. (5x − 2)2

345. (8x + y)2

Raising Binomials to the Third Power

346–350 Raise the binomials to the third power.

346. (x + 2)3

347. (y − 4)3

348. (3z + 2)3

349. (2x2 + 1)3

350. (a2 − b)3

Using Pascal's Triangle

351–360 Raise the binomial to the indicated power.

351. (x + 3)4

352. (y − 2)5

353. (z + 1)6

354. (a + b)7

355. (x − 2)7

356. (4z + 1)4

357. (3y − 2)5

358. (2x + 3)6

359. (3x + 2y)4

360. (2z − 3w)5

Finding Special Products of Binomials and Trinomials

361–365 Distribute the binomial over the trinomial to determine the “special” product.

361. (x − 1)(x2 + x + 1)

362. (y + 2)(y2 − 2y + 4)

363. (z − 4)(z2 + 4z + 16)

364. (3x − 2)(9x2 + 6x + 4)

365. (5z + 2w)(25z2 − 10zw + 4w2)

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