Algebra ( Period 5)

Factoring Trinomial Squares 8-9

Chapter 8-4 Déjà vu  Foil the following:

(a +3)² ( which is called a binomial squared)
(a + 3)(a +3) = a² + 3a + 3a + 9 = a² + 6a + 9 ( called a trinomial square)

Again, you see that the middle term is DOUBLE the product of the two terms in the binomial
and the first and last terms are simply the squares of each term in the binomial

How to recognize that it is a Binomial Squared:

1) Is it a trinomial? ( if it’s a binomial it cannot be a binomial squared)
2) Are the first  and last terms POSITIVE?
3) Are the first and last terms PERFECTSQUARES?4) Is the middle term DOUBLE the product of the SQRTS of the 1^st and last terms?

If YES to ALL of these questions you have a trinomial square

To FACTOR a Trinomial Squarea² + 6a + 9 (called a trinomial square)

1) Put ONE set of {{HUGS} with an exponent of 2  (  )²
2) Put the sign of the middle term inside  (in this case its +)  (  +  )²
3) Find the SQRT’s of the first term & last terms; place them in parentheses (a +3)²4) Check by FOILing back

If the sign is negative in the middle, simple use a negative sign when you factor.

Algebra (Period 5)

Using the Distributive Property 8-5

Using the Distributive Property to factor the GCF

(Your book also introduces “factoring by grouping” in this section but we will come back to that in a few days… We will also skip the ZERO PRODUCTS PROPERTY—for now)

Factoring is a skill that you MUST understand to be successful in high level math
Factoring is simply undoing multiplying.
Factoring uses the concept of DIVIDING
You are actually undoing the Distributive Property
How? You look for the most of every common factorà the GCF
Then you pull out the GCD (divide it out) from each term placing the GCF in front of a set of {{HUGS}}  parentheses

EXAMPLE:
Factor :  4m²n³ + 2m²n² + 6m²n
Step 1: What does each term have in common? (What is the GCF?)
They each can be divided by  2m²n  . Right ?
Step 2: Put the GCF in front of a set of {{HUGS}} and divide each term by the GCF

Step 3: Simplify and you get
2m²n(2n²+ n + 3)
Step 4: Check your answers!! Always check your factoring of the GCF by distributing back. (It should be the same thing!)

Another just as important check:
Make sure you have factored out the entire GCFà Look inside the parentheses and ask yourself:
Is there any common factors remaining between the terms?
If there is, you haven’t factored out the GREATEST Common Factor.

For example: Let’s say in the prior example that you only factored out 2mn

You would have:

 

Which would be    2mn( 2mn +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. Make sure you always look inside your {{HUGS}}

Relatively Prime Terms- terms with no common factors. That means that they cannot be factored ( GCF = 1)
We say they are “NOT Factorable” or “PRIME”