A General Factoring Strategy

I. GCF’s of Polynomials
A. The GCF for a polynomial is the monomial with the greatest coefficient and the lowest power on the variable (s)that will divide evenly into each term of the polynomial.
B. When we say factor the GCF of a polynomial, we mean to divide each term of the polynomial by the GCF and write the resulting answer as a product (times the GCF).

II. Factoring by Grouping
A. Procedure
1. Group terms that are similar together. Hint: We will usually group the terms in half, i.e. – If there are four terms, we’ll group them 2 by 2, 6 terms, 3 by 3, etc.
2. Factor out the GCF of each group.
3. Introduce a dummy variable (optional).
4. Factor out the GCF of the result.
5. Substitute back in for the dummy variable (optional, unless you did step 3, then it’s required).
6. Check by multiplying.

III. Factoring Trinomials with a Leading Coefficient of 1.
A. Factor out the GCF.
B. Set up two sets of parentheses.
C. Determine the factors of the First term.
D. Determine the signs in the parentheses:
1. If the last term is positive, the signs will be the same as the middle term:
a. ( + ) ( + )
b. (− ) ( − )
2. If the last term is negative, the signs will be positive and negative.

E. Determine the factors of the last term.
F. Check to ensure that you get the proper middle term. THIS IS NOT AN OPTIONAL STEP
G. This whole process is called the Trial and Error method.

IV. Factoring Trinomials with Leading Coefficients Other Than 1.
A. Factor out the GCF.
B. Set up two sets of parentheses.
C. Determine the factors of the First term.
D. Determine the signs in the parentheses:
1. If the last term is positive, the signs will be the same as the middle term:
a. ( + ) ( + )
b. (− ) ( − )
2. If the last term is negative, the signs will be positive and negative.

E. Determine the factors of the last term.
F. Check to ensure that you get the proper middle term.

V. Factoring Difference of Squares
A. Procedure for factoring Difference of Squares
1. Factor out the GCF
2. Set up two parentheses, where
a. The First numbers are the square root of the First term
b. The Last numbers are the square root of the Last term
c. The signs in between the first and the last terms are “+” in one parenthesis and “−“ in the other.
3. Check by multiplying.

VI. Sum and Difference of Cubes
A. This is a formula that you just have to learn, it is not at all intuitive.
B. Procedure
1. Factor out the GCF.
2. In the first set of parentheses, the First number will be the cube root of the First term
a. Take the cube root of the coefficient, but divide the exponent on the variable by 3 since perfect cubes will always have exponents that are multiples of 3.
3. Take the sign
a. “−“ if you have a difference of cubes
b. “+” if you have a sum of cubes
4. The Last number will be the cube root of the Last term.
5. In the second set of parentheses, the First number will be the square of the First number in the first set of parentheses.
6. The middle sign will be the opposite of the middle sign in the first set of parentheses.
7. Multiply the First and Last numbers from the first set of parentheses.
8. Add the square of the Last number from the first set of parentheses.
9. For those of you that like to see formulas, here they are:
a. F³ – L³ = (F – L)(F² + FL + L²)
b. F³ + L³ = (F + L)(F² - FL + L²)

VII. Perfect Square Trinomials
A. This is really just an extension of the Trial and Error method.
B. Remember that when a factor is multiplied by itself, we can say that it is squared.
C. For those that like to see formulas, they are:
1. F² + 2FL + L² = (F + L)²
2. F² − 2FL + L² = (F − L)²