Chapter 12

Other Factoring Techniques

The process of factoring binomials and quadratic trinomials is pretty much scripted with the various choices available for each format. When you start factoring expressions with more than three terms, you need different techniques to create the factorization — or to recognize that factors may not even exist.

The Problems You'll Work On

In this chapter on factoring polynomials, you deal with the following situations:

  • Factoring four, six, and eight (or more) terms, using grouping
  • Recognizing and using uneven grouping to create differences of squares
  • Starting with grouping and then finding differences of squares in the factors
  • Starting with grouping and then finding sums or differences of cubes in the factors
  • Recognizing that factoring out the GCF (greatest common factor) first makes factoring easier

What to Watch Out For

Here are a few things to keep in mind while you work on the factoring:

  • Making sure factors created from grouping are exactly the same
  • Factoring negative terms correctly
  • Recognizing when a problem can be factored again and knowing when to stop

Factoring by Grouping

536–543 Factor by grouping.

536. bc − 3b + 2c − 6

537. x2 − abx + xyz − abyz

538. 2x3 − 3x2 + 2x − 3

539. 2xz2 + 8x − 3z2 − 12

540. n3/2 + 2n − 4n1/2 − 8

541. y5/2 − 3y2 + 2y1/2 − 6

542. 4x − 12 + xy − 3y − xz + 3z

543. kx + 4x + ky + 4y + kz + 4z

Combining Other Factoring Techniques with Grouping

544–547 Factor each completely, beginning with grouping.

544. x2y2 + 3x2 − xy2 − 3x − 12y2 − 36

545. 2x4 − 4x2 + 3x3 − 6x + x2 − 2

546. m2n + 3m2 − 25n − 75

547. 4x3 + 16x2 − 25x − 100

Using Multiple Factoring Methods

548–565 Completely factor each expression.

548. 4x3 − 196x

549. 6x5 − 48x2

550. y5 − 4y3 − 27y2 + 108

551. x5 − 13x3 + 36x

552. 16x4 + 23x2 − 75

553. 4x6 − 4x2

554. z6 − 729

555. y8 − 1

556. 64b5 − 64b3 + b2 − 1

557. 27z5 − 243z3 − 8z2 + 72

558. z8 − 17z4 + 16

559. x5 − 2x4 + x3

560. x4 − 8x2 + 16

561. y−3 − 27y−6

562. (x2 − 1)2(3x + 4)2 + (3x + 4)3(x2 − 1)

563. (y3 + 8)4(y2 − 9)−(y2 − 9)2(y3 + 8)3

564. (z + 1)1/2(z3 − 1)2 −(z3 − 1)(z + 1)3/2

565. 4x6 − 25x4 + 500x3 − 3125x

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