Some
polynomials cannot be factored into the product of two binomials with
integer coefficients, (such as x^{2} +
16), and are referred to as
prime.
Other polynomials contain a multitude of factors.
"Factoring
completely"
means to continue factoring until no further factors can be found.
More specifically, it means to continue factoring until all factors
other than monomial factors are prime factors. You will have to
look at the problems very carefully to be sure that you have found all
of the possible factors.
To factor
completely:
1. Search for
a greatest common factor. If you find one, factor it
out of the polynomial.
2. Examine what remains,
looking for a trinomial or a binomial which can be factored.
3. Express the answer as
the product of all of the factors you have found. 


Factor completely:
1. 
Search for
the greatest common factor. In this problem, the greatest
common factor is 5.

2. 
Now, examine the
binomial x^{2}  9.
(Notice how the
factor of 5 is tagging along and remains as part of the answer.)

3.

Since the binomials (x  3) and (x
+ 3) cannot be factored further, we are done. Express the
answer as the product of all of the factors.

Factor
completely:
1. 
Search for
the greatest common factor. In this problem, the greatest
common factor is 4.

2. 
Now, examine and
factor the
trinomial x^{2}  6x  7.
Don't drop the 4.

3.

Since the binomials (x  7) and (x
+ 1) cannot be factored further, we are done. Express the
answer as the product of all of the factors.

