College Algebra - Unit 6

Simple Factoring

Group Factoring

AC- or FOIL Factoring

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What is Factoring?

Multiplying using distributivity

The Opposite Now!

Factoring Example

Factoring Example – leaving 1

Factoring out the GCF

• Thus when we have a set of terms and we want to factor them out first we look for the Greatest Common Factor

• Example:

Factor the following expression:

3x^2 + 6x = 3x(x+ 2)

Example

• Factor the following:

3xy^2 + 12xy

Example

• Factor the following:

3xy^2 + 12xy

3*x*y*y

3*2*2*y 3 x y are common!

3xy ( y + 4)

Group Factoring

Group Factoring

• Assume you have the following expression to factor:

3x + 3y + xa + ya

This expression has 4 terms.

STEP 1: We first split the terms into two groups

{3x + xa } and { 3y + ya}

when you group them choose the terms that have a common factor to put together

STEP 2: Factor each parenthesis

x( 3 + a) and y( 3 + a)

STEP 3: Now factor the parenthesis out from the two terms

(x+y)(3+a)

Factoring x^2 + bx + c

• To factor a polynomial like the above you need to find two numbers that if you multiply them, they give you c and when you add them they give you b.

• For example, if you have x^2 + 5x + 6 then you need to find two numbers p and q that their product is 6 and their sum is 5.

• Then x^2 + 5x + 6 = (x+p)(x+q)• Those numbers are 2 and 3 for this example.

Example x^2 + 11x + 30

• For example, if you have x^2 + 11x + 30 then you need to find two numbers p and q that their product is 30 and their sum is 11.

• Then x^2 + 11x + 30 = (x+p)(x+q)• Well, you can use trial and error search for those

numbers, or as I will show you next week you can follow a process to find those ;-)

• Those numbers are 5 and 6 for this example.• X^2 + 11x + 30 = (x+5)(x+6)

The FOIL ( AC) Method

• To factor now any polynomial ( trinomial ) of the form:

ax^2 + bx + c• We follow a method that is called Foil Method, or AC

method, depending the book you read.• It is not a difficult method, but it consists of 8 different

steps, if you follow those steps in the given order, you can factor almost all polynomials

• The group factoring that we discussed before is the last step of this method.

Foil Factoring

• Here we will start with an example on a general polynomial

Factor by grouping x^3 + 7x^2 + 2x + 14

First group the first two and last two terms.(x^3 + 7x^2) + (2x + 14)

Factor out the GCF from each binomial.X^2(x + 7) + 2(x + 7)

Write the GCF's as one factor and the common factor within the parentheses as the other factor.(x^2 + 2)(x + 7)

More complicated factoring example

To check the previous example:(x^2 + 2)(x + 7)= (x^2)(x) + (x^2)(7) + (2)(x) + (2)(7)= x^3 + 7x^2 + 2x + 14The product is the same as the original polynomial so the factors are correct

More complicated factoring example

Be careful with this!