Algebra (Period 1)

Difference of Two Squares 6-2

Again, remember that FACTORING just UNDOES multiplication.

In this case, the multiplication that you'll be UNDOING is FOILING.

FOIL:
(a + b)(a - b)

You will get:
a² - b²
This is the DIFFERENCE (subtraction) of TWO SQUARES.

Now FACTOR:
a² - b²
You undo the FOILING and get:
(a + b)(a - b)


REMEMBER:
You must have two different signs because that's how the MIDDLE TERM disappears!

You will get ADDITIVE INVERSES which will become ZERO



HOW TO RECOGNIZE THE DIFFERENCE OF TWO SQUARES:

1) Is it a binomial?

2) Is it a difference?

3) Are both terms perfect squares?


IF YES TO ALL 3 QUESTIONS, THEN YOU HAVE A DIFFERENCE OF 2 SQUARES!!

HOW TO FACTOR THE DIFFERENCE OF 2 SQUARES:

1) Double hug  (    )(    )

2) Find square root of each term (sq rt sqrt)(sq rt sq rt)

3) Make one sign positive and one sign negative.
              
(sq rt + sqrt)(sq rt -sq rt)


                           
Of course, they get more complicated! 
We can combine pulling out the GCF with this!

ALWAYS LOOK FOR A GCF TO PULL OUT FIRST!!!!!!

EXAMPLE:

27y² - 48y⁴ 

First, look for a GCF that can be pulled out.

The GCF = 3y²

Factor out the GCF (look at Chapter 6-1):
3y²(9 - 16y² )
NOW YOU HAVE A DIFFERENCE OF TWO SQUARES TO FACTOR:


3y²(3 - 4y)(3 + 4y)


CALLED FACTORING COMPLETELY BECAUSE
 YOU CANNOT FACTOR FURTHER!


Always check your factoring by distributing or FOILing back!


THERE IS NO SUCH THING AS THE SUM OF TWO SQUARES!

a² + b² CANNOT BE FACTORED!!!!!


BUT - b² + a²
= + a² -b²
= (a + b)(a - b)
BECAUSE IT'S JUST SWITCHED (COMMUTATIVE)