Some polynomials cannot be factored into the product of two binomials with integer
coefficients, (such as x2 + 16), and are referred to as prime,
while 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.  

detective
To factor completely:
 1.  Search for a greatest common factor.  If you find one, factor it out of the polynomial.

2. If the leading coefficient is negative, factor out the negative.

3.  Examine what remains, looking for a trinomial or a binomial which can be factored.
• if you have 2 terms, is it the difference of two squares?
• if you have 3 terms, is it a perfect square trinomial?
If not, can you form ( x _ __) (x _ __) binomial factors?

4.  Express the answer as the product of all of the factors you have found.

5. Check to see if your factors are correct.


ex1 Factor completely: 

1.
Search for the greatest common factor.  In this problem, the greatest common factor is 5.
2. Now, examine the binomial  x2 - 9.  This is the difference of two squares and can be factored. (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.

 

ex2 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  x2 - 6x - 7.  This pattern is not a perfect square trinomial, so go to two sets of parentheses. 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.

 


divider

NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
 and is not considered "fair use" for educators. Please read the "Terms of Use".