Factoring
Trinomials (a = 1)
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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 x, 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 +xYEA!!!!!
(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 x2  is 1,
then x2 + 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)

 

See how to use your
TI-83+/84+ graphing calculator  with factoring.
Click calculator.