A General Strategy for Factoring a Polynomial
1. Do all the terms in the polynomial have a common factor? If so, factor out the Greatest Common
Factor. Make sure that you don’t forget it in your final answer.
Next
2. Count the number of terms in the polynomial.
Two terms:
Is it a difference of squares, a2 - b2 ? Factor by using: a2 - b2 = (a - b)(a + b)
Is it a sum of squares, a2 + b2? The polynomial can’t be factored.
Three terms: Is it a perfect square trinomial? Factor by using: a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
Is it of the form x 2  bx  c ?
Factor by finding two integers that multiply to give c and add to give b. If the
integers are m and n, then x2 + bx + c = (x + m)(x + n).
Can’t find the integers? The polynomial can’t be factored.
Is it of the form ax2 + bx + c, a ≠ 1?
Factor using Trial and Error
or
Factor using the Decomposition Method. Find two integers that multiply to
give ac and add to give b. Use these integers to break up the middle term into
two terms. Factor the resulting four-term polynomial by grouping.
Can’t find the integers? The polynomial can’t be factored.
Four terms: Try Factoring by Grouping.
Group the 1st 2 terms and the last 2 terms to produce a common binomial
factor of the groups. Factor out the common binomial factor.
or
Group the 1st 3 terms or the last 3 terms. The 3 terms grouped form the terms
of a perfect square trinomial. The expression that results is a difference of
squares.
Remember
- Always factor completely. Double check that each of your factors can not be factored more.
- You can check your work by multiplying the factors together.
Example
Factor.
a) 8x3 - 6x2y2 + 4x2y
b) xy + 12 + 4x + 3y
c) -3x3 - 3x2 + 18x
d) 4x2 - 5xy - 6y2
e) y3 - 36y
f) 25x2 + 30xy + 9y2
g) 36m2 - 4n2 + 4n - 1