Unit 2, Lesson 3

Factoring

General Rules for Factoring

The first step in factoring any polynomial is to factor out the GCF (greatest common factor)

The next step is to determine how many terms there are in the polynomial. The number of terms will guide which strategy you use to factor

Factoring Polynomials with Two Terms- Binomials

There are really only a couple options for factoring a binomialThe first is if the binomial is a difference of squares a2 - b2 note that you might have to re-write the binomial to

have two “squares”

For example, x2 - 25 is a diff of squares x2 - 52

To factor it, remember that a2 - b2 = (a – b)(a + b)So, x2 - 25 = x2 - 52 = (x – 5)(x + 5)

Factoring Polynomials with Two Terms- Binomials

More examples:a2 – 49 = a2 – 72

(a – 7)(a + 7)y2 – 64 =

(y – 8)(y + 8)c2 – 9 =

(c – 3)(c + 3)4x2 – 100 =

(2x – 10)(2x + 10)

Factoring Polynomials with Two Terms- Binomials

The other option to factor a polynomial with two terms is if it is a difference or a sum of cubesWhile we won’t use this often, it does come up:a3 – b3 = (a – b)(a2 + ab + b2)a3 + b3 = (a + b)(a2 - ab + b2)

These formulas are on the inside page of your textbook and you will be able to use this page on quizzes and tests from now on

Factoring Polynomials with Three Terms- Trinomials

This was covered in the last lesson –The easiest trinomial to factor is a PST (perfect square trinomial) so always check for this firstPSTs have first and third terms that are squares and the middle term is two times the roots of the first and third termsEx. x2 + 10x + 25 = (x + 5)2

Ex. t2 - 16t + 64 = (t - 5)2

Factoring Polynomials with Three Terms- Trinomials

Another way to determine if a trinomial is a PST is to take half of the middle term coefficient and square it. If your answer is the third term, it’s a PSTEx. x2 + 6x + 9 ½ of 6 is 3; 32 = 9 so it is a PSTWhat would the third term have to be for the following to be PSTs?x2 + 18x + ?

81x2 + 12x + ?

36x2 + 4x + ?

4

Factoring Polynomials with Three Terms- Trinomials

If a trinomial is not a PST, you are left to do some “guess and check” workRemember that multiplying two binomials often times yields a trinomial, so you can guess that your answer will be two binomials (if the trinomial can be factored)Ex. x2 + 11x + 24 this is not a PST

The answer will look like (x + )(x + )To find the second terms in each binomial, you need to find two numbers that when multiplied give 24 and when added are 11So: x2 + 11x + 24 = (x + 8)(x + 3)

Factoring Polynomials with Three Terms- Trinomials

A couple of things to keep in mind:• If the third term is negative, then one of the

binomials uses subtraction and one uses addition Why?

Ex. x2 + x – 6 = (x + 3)(x – 2) Ex. x2 - 2x – 8 = (x + 2)(x – 4) • Not all trinomials can be factored

Factoring Polynomials with Four terms or More

The key here is to look at the terms in groups of twos or threes. Then, use the factoring strategies you already know:1. GCF2. Two terms- diff of squares; sum/diff of cubes3. Three terms- PST or “two parentheses”

method

Factoring Polynomials with Four terms or More

Ex. 3x3 + 5x2 – 6x -10To begin, factor just the first two terms:3x3 + 5x2 the GCF is x2 so: x2 (3x + 5)Next, factor the second two terms:– 6x -10 the GCF is -2 so: -2(3x + 5)Notice that you get (3x + 5) in both cases, and so that is one factor and the other factor is (x2 – 2)So, 3x3 + 5x2 – 6x -10 = (3x + 5)(x2 – 2)

Factoring Polynomials with Four terms or More

Last example: 4x3 – 8x2 + 3x – 6Take the GCF of the first two terms:4x2(x – 2)Take the GCF of the second two terms:3(x – 2)Use (x – 2) as one factor and (4x2 + 3) as the other factor4x3 – 8x2 + 3x – 6 = (x – 2)(4x2 + 3)