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Factoring

•Factoring out a Common Factor : The first step in factoring any polynomial is to
look for anything that all the terms have in common and then factor it out using the
distributive property .

Example: Here, the terms share the common factor 5y2 (i.e. 5 is the largest
number that divides both 20 and 5, and both terms contain the variable y with 2 being
the smallest exponent). So we factor it out:

•Factoring by Grouping: Factoring by grouping is useful when we encounter a polynomial
with more than 3 terms.
Example:

1. First, we group together terms that share a common factor.
The first group shares an x2 and the second shares a -6.

2. Factor out the common factor from each grouping. You should have left the same
expression in each group . Here that expression is 3x+1
3. Now factor out that expression.

Factoring Trinomials - Reverse FOIL: There two basic cases that we’ll encounter:

1. The leading coefficient is a 1. This is the easier of the two cases : x2 + bx + c All
we need to do here is find two numbers whose product is c and sum is b
Example
: x We need to find two numbers that
multiply to give us +10, but add to give us -7. Well, -5 and -2 do the trick. So


2. The leading coefficient is not a 1. Things are a little trickier here, but not much.
Again, it’s just FOIL in reverse.
Example:
We need two numbers to fill in for the hearts that will multiply to 3. How about
3 and 1?

Now we need two numbers to fill in for the triangles that will multiply to -20
AND when we do the INNERS and OUTERS we get 7y. We’ll use the GUESS
and CHECK method to find the two numbers we need.
Let’s try 10 and -2 first:

That’s not it! Maybe 5 and -4?

Close, but the sign on the 7 is wrong . Easy to fix - just switch the signs on the 5
and 4:
Presto !!

• Special Factorizations: Some polynomials are easy to factor because they fit a
certain mold.

Difference of Squares :
Example:

Perfect Squares : These are polynomials that factor into
The pattern we’re looking for here is
Example:
Example:
Difference of Cubes :
Example:
Sum of Cubes: )
Example:

•Strategy for Factoring:
1. Always factor out the largest common factor first. This will make life easier for
any further factoring that may need to be done.

2. Look at the number of terms
– Two terms: Is it a difference of squares, difference of cubes or sum of cubes?
– Three terms: Is it a perfect square? Try reverse FOIL.
– Four or more terms: Try factoring by grouping.

3. Always make sure the polynomial is factored COMPLETELY.

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