Algebra (Period 1)

Factoring Polynomials 6-1


Chapter 5 was a very important building block of Algebra but

CHAPTER 6 IS EVEN MORE IMPORTANT FOR HIGH SCHOOL!!!



REMEMBER THIS KEY CONCEPT:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL …SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(You'll either distribute or FOIL.)


Factoring is a skill that you must understand to be successful in higher level math!!!

We did a simple version of this back in Chapter 1 and you had a RACE on it!


Factoring is simply UNDOING multiplying


Say you multiplied 5 by 10 and got 50

How would you undo it?
DIVIDE by 5!


So FACTORING uses the concept of DIVIDING.

You're actually undoing the DISTRIBUTIVE PROPERTY.

How?
You look for the most of every common factor....the GCF!

Then you pull out the GCF (divide it out of) from each term,
placing the GCF in front of ( )


EXAMPLE:

FIRST, DISTRIBUTE:
2m²n (2n2 + n + 3)
=
4m²n³ + 2m²n² + 6m²n


Now, pretend you don't want the 2m²n to be distributed anymore...

What should you end up with once you UNDO the Distributive Property?

2m²n (2n² + n + 3)


That's exactly what you started with!

So is it that easy?

Well yes... and no...

Yes because that is the answer
…and
…
No because it was only that easy because I gave you how it started!

You won't know how it started in a real problem!



THIS IS AN EXAMPLE OF THE WAY YOU WOULD USUALLY SEE IT.

The question would say:
FACTOR:
 4m²n³ + 2m²n² + 6m²n


Step 1: What does each term have in common (what is the GCF) ?

They each can be divided by 2m²n


Step 2: Put the GCF in front of a set of ( ) and divide each term by the GCF
2m²n ( 4m²n/2m²n + 2m²n²/2m²n + 6m²n/2m²n)


Step 3: SIMPLIFY and you'll get:

2m²n (2n² + n + 3)



Step 4: Check your answer!!!!!

Always check your factoring of the GCF by distributing back!


(incognito, it should be the same thing)

Another check is to make sure you have factored out the entire GCF.

Look inside the parentheses and ask yourself if there is still any factors in common between the terms.
If there is, then you haven't factored out the GREATEST Common Factor.
For example, let's say in the prior example that you only factored out 2mn instead of 2m²n. You would have:
2mn ( 4m²n³/2mn + 2m²n²/2mn + 6m²n/2mn)


= 2mn(2mn²+ mn + 3m)


If you just check by distributing back, the problem will check.

BUT ...
Look inside the ( ) and notice that each term still has a common factor of m!
So this would not be the fully factored form!
So make sure you always look inside the ( )!!!

RELATIVELY PRIME TERMS -

TERMS WITH NO COMMON FACTORS

THAT MEANS THAT THEY CANNOT BE FACTORED
(GCF = 1)

We say they are "not factorable"