Chapter 11

Factoring Quadratic Trinomials

You can factor trinomials with the form ax2 + bx + c in one of two ways: (1) factor out a GCF, or (2) find two binomials whose product is that trinomial. When finding the two binomials whose product is a particular trinomial, you work from the factors of the constant term and the factors of the coefficient of the lead term to create a sum or difference that matches the coefficient of the middle term. This technique can be expanded to trinomials that have the same general format but with exponents that are multiples of the basic trinomial.

The Problems You'll Work On

Here are the types of things you work on in this chapter:

  • Factoring out a GCF (greatest common factor)
  • Creating the product of two binomials, both with variable coefficients of 1
  • Creating the product of two binomials, one or both with variable coefficients not equal to 1
  • Applying the techniques to quadratic-like trinomials
  • Using more than one factorization method in a problem

What to Watch Out For

Be aware of the following when factoring quadratic trinomials:

  • Assigning the correct sign to each term, especially when a factor or term is negative
  • Positioning the signs correctly in the product of binomials so a difference has the correct sign after cross-multiplying
  • Finding the correct factors of coefficients and constants when you have several to choose from in the problem
  • Recognizing when a factor in a problem can be factored again

Factoring Out the GCF of a Trinomial

496–499 Factor out the GCF of each.

496. 12x4y2 − 6x3y3 + 21x2y4

497. 70a2b3c + 63a3b2c2 − 21a4bc3

498. 3(x − 4)3 + 6x(x − 4)2 − 9x2(x − 4)

499. 60x5y − 48x6y2 + 36x2y3

Factoring Trinomials into the Products of Binomials

500–511 Factor each trinomial into the product of two binomials.

500. x2 − 8x − 20

501. x2 + 10x + 9

502. y2 − 6y − 16

503. z2 + 2z − 48

504. 2x2 + x − 6

505. 3x2 + 5x − 12

506. 9z2 + 24z + 16

507. 16x2 − 40x + 25

508. w2 − 63w − 64

509. 4x2 + 15x − 25

510. 40x2 − 3x − 54

511. 16x2 − 14x − 15

Factoring Quadratic-Like Expressions

512–519 Factor the quadratic-like expressions into the product of two binomials.

512. x10 − 5x5 + 4

513. y6 − 4y3 − 21

514. y16 − 25

515. 25a4 − 49b10

516. x−8 − 3x−4 − 18

517. x−6 + 5x−3 + 4

518. 5x1/3 − 11x1/6 + 2

519. 6x2/5 − x1/5 − 12

Factoring Completely Using More Than One Technique

520–535 Completely factor each trinomial.

520. 5z2 + 30z + 45

521. 18x3 + 12x2 + 2x

522. 4y3 − 8y2 − 12y

523. 6x6 + x5 − x4

524. x5 + 8x4 + 16x3

525. 96y − 48y2 + 6y3

526. w4 − 13w2 + 36

527. x6 − 9x3 + 8

528. 5x2(x + 3)3 + 15x(x + 3)3 − 50(x + 3)3

529. 4x2(x + 1) − 6x(x + 1) − 4(x + 1)

530. a2(x2 − 81) + 13a(x2 − 81) + 22(x2 − 81)

531. 4y2(x3 − 1) + 4y(x3 − 1) −3(x3 − 1)

532. 300y1/4 + 70y1/8 − 150

533. 6y − 6y1/2 − 12

534. 3x−3 − 19x−2 + 20x−1

535. 12x−2 + 5x−3 − 2x−4

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