How to factor polynomials

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The methods that are used to factor a polynomial depend on how many terms the polynomial has.
Note: This page assumes that the polynomials are being factored on the real field (that the answers are real numbers).


Any Polynomial


The first step to factor any polynomial is always to factor out the Greatest Common Factor, commonly referred to as the GCF.

For example:

 


or

 


or

 



Binomial (2 Terms)


Again, the first step is to factor out the GCF. If there is no GCF, then there are only 3 possibilities: Difference of Squares, Sum of Cubes, or Difference of Cubes.

Difference of Squares

 

For example:
 

or

 

Sum of Cubes

 

For example:
 

or

 



Difference of Cubes

 

For example:
 

or

 



Trinomial (3 Terms)

There are three possibilities for factoring a trinomial depending on which type of trinomial it is.


Monic Trinomials

A monic trinomial has 1 as the leading coefficient.
       where   and  .

For example:
  because   and  

or

  because   and  


Non-Monic Trinomials

A non-monic trinomial has a constant other than 1 as the leading coefficient.

 
where  ,  , and  

Many times students are taught that to factor a non-monic trinomials, they must guess different combinations of m,n,p,and q and then FOIL the factors to see if they had guessed correctly. There is a method of factoring that, while not often taught, will work.

Example:
Factor  

Step 1   Multiply a and c. (Multiply the number in front of   and the constant)
Multipy 6 and -3,  

Step 2   Find factors of ac.
Find factors of -18: -1(18), 1(-18), -2(9), 2(-9), -3(6), and 3(-6).

Step 3   Decide which factors of ac that when added together will give b.
The combination of -2 and 9 is the one needed since -2+9=7.

Step 4   Rewrite the middle term of bx using the factors found in step 3.
Instead of  , write  

Step 5   Factor by grouping.
 
   

Therefore  

Perfect Square Trinomials

Perfect square trinomials are of the form   or  
  and
 

For example:
 

or

 


Polynomials with 4 terms

Polynomials with 4 terms can be factored by some form of grouping. There are special groupings but the most common form is referred to as factoring by grouping and is described step by step below.

Example:
 

Step 1   Split the polynomial into groups of 2 terms.
 

Step 2   Find the GCF (greatest common factor) of each group.
 

Step 3   If the 'leftovers match' factor them out.
Since there is a (a-3b) in each term, factor out (a-3b) from each term.
 

Therefore  


When nothing works

If the polynomial can't be factored, then it is considered prime.

References

Lial, Margaret L.; Hornsby, John; McGinnis, Terry (2008). Intermediate Algebra 10th edition. Boston: Addison Wesley. pp. 346–388. ISBN 978-0321443625.

Tussy, Alan S.; Gustafson, R. David (2008). Intermediate Algebra 4th edition. Brooks Cole. ISBN 978-0495389736. {{cite book}}: Unknown parameter |month= ignored (help)