Algebra Period 4

Factoring ax² +bx + c Section: 6-5

Checklist of how to factor thus far:

1. Look for a GCF of all terms

2. Binomials - look for difference of two squares
 both perfect squares - double hug - one pos, one neg - square roots of both terms

3. Trinomials - look for Trinomial Square (factors as a binomial squared)

first and last must be perfect squares - middle must be double the product of the two square roots

SINGLE hug - square roots of both terms - sign is middle sign

4. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle

5. Trinomials - last sign negative - double hug with different signs,
putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor

6. 4 terms - Factor by grouping - make sure in descending order - pair off the first 2 terms and the last 2 terms -
make sure there's a PLUS sign in between the 2 pairs -
factor out the GCF of each pair - if the binomial left in the (  ) is the same, it's factorable.


REMEMBER:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!

Factoring ax²+ bx + c  Sections 6-5 or
FACTORING TRINOMIALS WITH A COEFFICIENT ON THE 1ST TERM.  
We'll use FACTORING BY GROUPING (Chapter 6-6)

I call this "Xbox 360" and you'll see why in class!
When you have a trinomial with a coefficient on the first term, factoring becomes more difficult.

A good method to factor is to use factoring by grouping.

1) Multiply the first term's coefficient by the last term's coefficient

2) By guess and check, find 2 factors that multiply to the product you got in #1 and add to your middle term
(just like how you did it for trinomials without a coefficient on the first term)
I like to set up a T Chart and go from there!!

3) Rewrite the original trinomial as a 4 term polynomial using the first term, the two factors you found, and the the last term.
IT SHOULD SIMPLIFY TO THE ORIGINAL PROBLEM!

4) Factor by grouping

AGAIN, NOT ALL TRINOMIALS ARE FACTORABLE!


EXAMPLE: 15n² - 19n - 10

1) Multiply 15 by 10 = 150

2) Find 2 factors that multiply to 150 and add to -19 (which means subtract to 19)

Think: 15 and 10? NO

30 and 5? NO

25 and 6? YES! 

Since the negatives must win, it must be -25 and +6 = -19!


3) Rewrite 15n² - 19n - 10 as a 4 term polynomial:
15n² - 25n + 6n - 10


4) Factor by grouping: (15n² - 25n) + (6n - 10)

5) Pull out the GCF: 5n(3n - 5) + 2(3n - 5)

(3n - 5)(5n + 2)


6) Check to make sure you can't factor any more.

7) FOIL to make sure you did not make a silly mistake!