Algebra Honors

Factoring the Difference of Two Squares  8-8

Again… remember that FACTORING just UNDOES multiplication

Chapter 8-3 Déjà vu  Foil the following:
(a + b)(a –b) you will get a² – b²

This is the DIFFERENCE 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 DIFFERENCE of TWO SQUARES

1) Is it a binomial?
2) Is it a difference?
3) Are both terms perfect squares?

If YES to all three questionsàthen you have DIFFERENCE of TWO SQUARES

How to factor the DIFFERENCE of TWO SQUARES

1) Put a set of double {{HUGS}}  (      )(       )
2) Find the square root of each term  (SQRT   SQRT) (SQRT   SQRT)
3) Make one sign positive and one sign negative  (SQRT+SQRT) (SQRT-SQRT)

Of course, they can get more complicated
Always look for a GCF FIRST to pull out

EXAMPLE:

27y² – 48y⁴

At first this just looks like a binomial ànot the difference of two squares…
The GCF is 3y²
3y²(9 – 16y²)
WOW—Now we have the DIFFERENCE of TWO SQUARES
3y²(3 + 4y)(3 - 4y)

Called FACTORING COMPLETELY because you cannot factor any further
Always check your factoring by distributing or FOILing back!BTW…  There is NO SUCH THING as the Sum of Two Squares!

a² + b² CANNOT be factored!!

but… -b² + a² is really a² –b² because it is just switched ( commutative)