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

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 + 14 The product is the same as the original polynomial so the factors are correct More complicated factoring example

Be careful with this!