Chapter 10

Factoring Binomials

A binomial is an expression with two terms. The terms can be separated by addition or subtraction. You have four possibilities for factoring binomials: (1) factor out a greatest common factor, (2) factor as the difference of perfect squares, (3) factor as the difference of perfect cubes, and (4) factor as the sum of perfect cubes. If one of these methods doesn't work, the binomial doesn't factor when using real numbers.

The Problems You'll Work On

The problems in this chapter focus on the following:

  • Factoring when the two terms are the difference of perfect squares (both the numbers and the variables must be perfect squares)
  • Factoring when the two terms are the difference of perfect cubes (both the numbers and the variables must be perfect cubes)
  • Factoring when the two terms are the sum of perfect cubes (both the numbers and the variables must be perfect cubes)
  • Using more than one factorization technique in a problem

What to Watch Out For

When working through the steps necessary for factoring binomials, pay careful attention to the following:

  • Recognizing when a number is a perfect square so you can apply the factorization technique
  • Knowing enough of the perfect cubes to recognize them in binomials
  • Using the correct sign between the first and second terms of the trinomial when factoring sums and differences of cubes
  • Trying to factor the sum of perfect squares and mistaking it for the technique used with cubes
  • Using the correct exponents when factoring higher-power squares and cubes and dividing the exponents, not taking their root

Factoring the Difference of Perfect Squares

456–465 Factor each binomial using the pattern for the difference of squares.

456. x2 − 36

457. 9y2 − 100

458. 81a2 − y2

459. 4x2 − 49z2

460. 64x2y2 − 25z2w4

461. 36a4b6 − 121

462. 121x1/2−144y1/4

463. 25x−2 − 9y−4

464. 16 − x2y−1/4

465. z−4/9 − 49w1/2

Factoring the Sum or Difference of Two Perfect Cubes

466–475 Factor each as the sum or difference of perfect cubes.

466. x3 + 8

467. x3 + 343

468. a3 − 216z3

469. 1 − y3

470. 125z3 + 343

471. 8a3 + 27b3

472. 729x3 − 1000y6

473. 512x9 − 125y27

474. 27x1/3 − 1

475. 8y−6 + 343z−1

Factoring More Than Once

476–495 Completely factor each binomial.

476. 3x4y3 − 75x2y3

477. 6x4y2 − 96x2y4

478. 36z2 − 3600w2

479. 100x3 − 900x

480. 32y4 + 4y

481. 4x4y2 + 32xy2

482. 625x4 − 1

483. 16x4 − 81y8

484. x−4 − x−7

485. y−8 − y−12

486. 216a3b3 + 216c6

487. 125a2b4 − 500c6

488. a3b6 − 8b6

489. 81x2y3 + 3x2

490. 32x4y − 4xy

491. 9a6b5 + 72z3b2

492. 2x6 − 162x2

493. x6 − 1

494. y12 − 64

495. 10a4x2 − 1000b8x2

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