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: 20y2 - 5y5 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: 20y2 - 5y5 = 5y2(4 - y3)
Factoring by Grouping: Factoring by grouping is useful when we encounter a polynomial
with more than 3 terms.
Example: 3x3 + x2 - 18x - 6
1. First, we group together terms that share a common factor . (3x3 + x2) + (-18x - 6)
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 . x2(3x+1)+(-6)(3x+1) Here that expression is 3x+1
3. Now factor out that expression. (3x + 1)(x2 - 6)
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: x2 - 7x + 10 = (x + △)(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
x2 - 7x + 10 = (x + (-2))(x + (-5)) = (x - 2)(x - 5)
2. The leading coefficient is not a 1. Things are a little trickier here, but not much.
Again, it’s just FOIL in reverse.
We need two numbers to fill in for the hearts that will multiply to 3. How about
3 and 1?
3y2 + 7y - 20 = (3y +△)(1y + △)
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:
(3y - 2)(y + 10) = 3y2 + 30y - 2y - 20 = 3y2 + 27y - 20
That’s not it! Maybe 5 and -4?
(3y + 5)(y - 4) = 3y2 - 12y + 5y - 20 = 3y2 - 7y - 20
Close, but the sign on the 7 is wrong . Easy to fix - just switch the signs on the 5
(3y - 5)(y + 4) = 3y2 + 12y - 5y - 20 = 3y2 + 7y - 20 Presto!!
Special Factorizations: Some polynomials are easy to factor because they fit a
– Difference of Squares : F2 - L2 = (F + L)(F - L)
Example: 16x2 - 9 = 42x2 - 32 = (4x)2 - 32 = (4x + 3)(4x - 3)
– Perfect Squares : These are polynomials that factor into (F + L)2 or (F - L)2
The pattern we’re looking for here is F2 + 2LF + L2 or F2 - 2LF + L2
Example: x2 + 6x + 9 = x2 + 2·3x + 32 = (x + 3)2
Example: y2 - 10y + 25 = y2 - 2·5y + 52
– Difference of Cubes : F3 - L3 = (F - L)(F2 + LF + L2)
Example: 2z3 - 54 = 2(z3 - 27) = 2(z3 - 33) = 2(z - 3)(z2 + 3z + 9)
– Sum of Cubes: F3 + L3 = (F + L)(F2 - LF + L2)
Example: n3 + 216 = n3 + 63 = (n + 6)(n2 - 6n + 36)
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.