Factoring and Quadratic Equations

Here we will be factoring special binomials. We have seen some of the formulas in a
previous section but we will be using them in a different manner here. These formulas
will be used as a template for factoring.


does not factor
Difference of Squares

does not factor - Sum of Squares

- Difference of Cubes

- Sum of Cubes

The idea is to identify the a and the b then plug in to the formulas above

   
Step 1: Identify the special
binomial.
No need to check but it
doesn’t hurt.
Step 1: Identify the special
binomial.
  Step 3: Plug into the formula.

The same steps can be used for the sum and difference of cubes formula.

 

Difference of cubes

At this point it is nice to review the perfect cubes that we will be likely to run into when
working problems in this section.
and . Many times the coefficients will give a clue as to what special binomial
formula is to be used.

A. Factor






At this point we will look a bit more closely at the process in which we are factoring sum
of cubes using the formula

Once we identify the a and the b in the formula it really is just a matter of plugging in to
the formula or using it as a template. Most of this is done mentally, so it is sufficient to
present your solution without the above steps.

Rest assured that with much practice you will be able to jump straight to the answer too.
The first step to this ability, of course, is to memorize the formulas.





As we have seen before, we will often run into polynomials with a GCF. It is important
with special binomials to factor out the GCF first.

   
Doesn’t really look like difference of squares. Factor out the GCF and it is easier to see.
Difference of squares

B. Factor





 
Sum of squares does not factor for us.

Factoring binomials is a bit more complicated when larger exponents are involved. It is
difficult to recognize that x6, for example, is a perfect cube. We can think of
or the cube of x squared. Also, recall the rule of exponents

   
Sum of cubes
Here and
in the formula

   

It is not always necessary to show all the steps shown above. Ask your instructor what he
or she wants to see in the way of steps when presenting your solutions in this case.

C. Factor






 
Factor using difference of squares again.

Tip: Look out for problems that require us to factor the factors. In other words, look to
continue factoring until all factors are completely factored. Also, the trinomials that we
get when using the sum and difference of squares will not factor for us so do not even try.



 
ALWAYS factor difference
of squares first


If we are confronted with a polynomial that is both a difference of squares and a
difference of cubes we must factor it as a difference of squares first. Doing this will
ensure that our formulas will achieve the best possible factorization.