© William James Calhoun, 2001

10-5: Perfect Squares and Factoring

OBJECTIVE:You will identify and factor perfect square trinomials.

These two tools are the reverse of the perfect square tools from last chapter.

These trinomials can still be factored using trinomial factoring.

If you recognize the trinomial as one of the perfect squares, use the tools from this section.

If you do not recognize the trinomials as one of the perfect squares, you just use trinomial factoring.

© William James Calhoun, 2001

10-5: Perfect Squares and Factoring

10.5.1 PERFECT SQUARE TRINOMAILS

a2 + 2ab + b2 = (a + b)2

a2 - 2ab + b2 = (a - b)2

EXAMPLE 1: Determine whether each trinomial is a perfect square trinomial. If so, factor it.A. 4y2 + 36yz + 81z2 B. 9n2 - 21n + 49Is the first term a perfect square?

yes, 2yIs the last term a perfect square?

yes, 9zIs the middle term 2 times 2y times 9z

yes, 2(2y)(9z) = 36yzThis is a positive perfect square, use 10.5.1.

(2y + 9z)2

Is the first term a perfect square?yes, 3n

Is the last term a perfect square?yes, 7

Is the middle term 2 times 2y times 9zno, 2(3n)(7) = 42n

This is not a perfect square trinomial.You should try trinomial factoring, but the instructions say stop if it is not a perfect square.

© William James Calhoun, 2001

10-5: Perfect Squares and Factoring

The final stage of factoring is taking any polynomial and deciding which factoring method to use.Follow these steps:

1) Pull out the GCF.2) If there are four terms, use grouping to factor.3) If there are two terms, check for differences of squares.

(Remember sum of squares cannot be factored.)

4) If there are three terms, you can check for perfect squares.5) Use trinomial factoring.6) Go through this list again to look for factors that can be factored

further.

Remember that you might need to rearrange terms every now and then.

© William James Calhoun, 2001

10-5: Perfect Squares and Factoring

EXAMPLE 2: Factor each polynomial.A. 4k2 - 100 B. 9x2 - 3x - 20

4(k2 - 25)4(k - 5)(k + 5)

C. 4m4n + 6m3n - 16m2n2 - 24mn2

9x2 - 15x + 12x - 203x(3x - 5) + 4(3x - 5)(3x - 5)(3x + 4)

2mn(2m3 + 3m2 - 8mn - 12n)

2mn[m2(2m + 3) - 4n(2m + 3)]2mn[(2m + 3)(m2 - 4n)]2mn(2m + 3)(m2 - 4n)

© William James Calhoun, 2001

10-5: Perfect Squares and Factoring

HOMEWORK

Page 584#23 - 41 odd

Page 591#21 - 39 odd