Question 138: How to you factor trinomials ? Need to be explained
Thanks alot ~!~
Answer by KMST(5484)   (Show Source):
You can put this solution on YOUR website! STRATEGIES TO FACTOR TRINOMIALS:
If your trinomial looks like  where  and  are integers, you start by looking for factors of ,
because if you are going to find factors  and  whose product is the trinomial,
you need to find  and  .
In that case,  for every value of  ,
so and must be such that  and 
If you have  , you look for and such that  , and  .
To start, you need factors whose product is  .
Those are:
 ,
 , and
 .
Then, you will have to give one of the factors a minus sign, to make the product .
The factors (with their signs) have to add up to .
and  do not work,
because  , but
 work, so  and  are your and .
It does not matter which one is , or which one is in the first factor.
 or  is your factoring, anyway you look at it.
What if there is a coefficient before  , like  ?
So, let's say that your trinomial is  .
That changes things a bit.
You could say the factoring could be 
  .
Then,  -->  ,
and you want to factor  to begin,
to find factors  and  such than  .
Let me show you an example:
Let's say I want to factor  , with  ,  , and  ,
We calculate  , and look for factor and such that  .
I should start by finding factors of  ,
but I will find the factor pairs with a product of  first,
and figure positive or negative signs later.
 , but that does not work, because  ,
 , but that does not work, because  ,
 , but that does not work, because  ,
 , and that works, because  ,
so , and are  , and  .
I write the trinomial  ,
from what I found before, as,
 and factor once in pairs,
and then factor once more to get the full factorization:
IN GENERAL:
A trinomial is an expression that is the sum of 3 terms.
In math, when we talk about factoring trinomials,
we usually mean factoring polynomials of degree 2 with 3 terms, like  .
CAUTION:
If you are asked to factor a trinomial in algebra class,
it is likely that it will be possible and not too difficult,
but not all degree 2 trinomials can be factored.
Some are the product of 2 nice looking binomial factors.
For example,  is the product of two binomial factors.
However, when we are working with only real numbers,
not all polynomials of degree 2 with 3 terms can be factored.
Some cannot be factored into factors with real coefficients.
Other can be written as a product of two factors including real numbers as coefficients, but those factors do not look nice,
as in  .
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