Note: NYS Integrated
Algebra requires the factoring of only trinomials whose leading
coefficients are 1.
If you need to deal with leading coefficients other than one, go to
Algebra 2.
for this lesson
a will
always be 1.

Whether
we use the distributive process, use FOIL, or line up the factors vertically to multiply,
we all know that:
The
expression
is called a
quadratic trinomial.
To factor a trinomial of this form, we need to reverse the multiplication
process we used above.
ATTENTION
Super Sleuths:
We are on the hunt for factors!
There are many different ways to think about this process of "reversing"
multiplication. Let's look first at what is happening
and then at a shortcut process for finding the factors.



Let's see what
is involved with factoring
.
1. 
To get the
leading term of x², each first term will be x. So we start
with:
(x
) (x )

2. 
The product of the
last terms must be 6. Unfortunately, we are now faced
with options, as there are several ways to arrive at a product
of 6:
+6 and 1
6 and +1
+3 and 2
3 and +2 
All of these options will give us a product of 6. 
These different options make it "appear" that
we have several possible answers:
(x + 6)(x  1)
(x  6)(x + 1) (x +
3)(x  2)
(x  3)(x + 2) 


3.

The possible answers created from our options
above do not ALL give us the correct result. We need to
find the combination that will yield
the correct "middle term" of +x (for this
problem).
(x + 6)(x  1) gives
middle term 5x.
(x  6)(x + 1) gives
middle term 5x.
(x + 3)(x  2) gives
middle term +x. YEA!!!!!
(x  3)(x + 2) gives
middle term x.

4. 
Answer:
Notice: While we initially had
several options for answers, we really had only one true answer.
The more options that a problem creates, the more detective work needed to find
the true answer. 
SHORT CUT
PROCESS:
If the
coefficient of x^{2} is 1,
then x^{2 }+ bx + c = (x + m)(x
+ n)
where m and n multiply to give c
and m and n add to give b. 


When the leading
coefficient is 1,
ask yourself
"what numbers multiply to the last term
and add to the middle term?"
In the example above,
,
you need numbers that multiply to 6 and add to +1.
These numbers will be +3 and 2 and create and answer of: (x
+ 3) (x  2)
