Factoring Polynomials

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§ 13.5

Factoring Perfect Square Trinomials and the Difference of Two Squares

Slide 44

Perfect Square Trinomials

Recall that in our very first example in Section 4.3 we attempted to factor the polynomial 25x2 + 20x + 4.

The result was (5x + 2)2, an example of a binomial squared.

Any trinomial that factors into a single binomial squared is called a perfect square trinomial.

Slide 45

In the last chapter we learned a shortcut for squaring a binomial

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

(a – b)2 = a2 – 2ab + b2

So if the first and last terms of our polynomial to be factored are can be written as expressions squared, and the middle term of our polynomial is twice the product of those two expressions, then we can use these two previous equations to easily factor the polynomial.

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

a2 – 2ab + b2 = (a – b)2

Perfect Square Trinomials

Slide 46

Factor the polynomial 16x2 – 8xy + y2.

Since the first term, 16x2, can be written as (4x)2, and the last term, y2 is obviously a square, we check the middle term.

8xy = 2(4x)(y) (twice the product of the expressions that are squared to get the first and last terms of the polynomial)

Therefore 16x2 – 8xy + y2 = (4x – y)2.

Note: You can use FOIL method to verify that the factorization for the polynomial is accurate.

Perfect Square Trinomials

Example

Slide 47

Difference of Two Squares

Another shortcut for factoring a trinomial is when we want to factor the difference of two squares.

a2 – b2 = (a + b)(a – b)

A binomial is the difference of two square if

both terms are squares and

the signs of the terms are different.

9x2 – 25y2

– c4 + d4

Slide 48

Difference of Two Squares

Example

Factor the polynomial x2 – 9.

The first term is a square and the last term, 9, can be written as 32. The signs of each term are different, so we have the difference of two squares

Therefore x2 – 9 = (x – 3)(x + 3).

Note: You can use FOIL method to verify that the factorization for the polynomial is accurate.

Slide 49

§ 13.6

Solving Quadratic Equations by Factoring

Slide 50

Zero Factor Theorem

Quadratic Equations

Can be written in the form ax2 + bx + c = 0.