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East Campus, CB 117
361-698-1579
Math Learning Center
West Campus, HS1 203
361-698-1860
FACTORING SUMMARY
Factor out
GCF
2 terms
3 terms
(binomial)
(trinomial)
ax2+bx+c
4 terms
(polynomial)
Difference of
Squares
(__+__)(__-__)
Sum of Squares
Trial and Error
(__ __)(__ __)
PRIME
a=1
a≠1
Difference of Cubes
(_ – _)(_ + _ + _ )
Sum of Cubes
(_ + _)(_ – _ + _ )
AC Method
by Grouping
East Campus, CB 117
361-698-1579
Math Learning Center
West Campus, HS1 203
361-698-1860
Factoring Summary
I.
Factor out the Greatest Common Factor (GCF) first.
A. Find the largest number that divides evenly into all of the coefficients.
B. If the first coefficient is negative, factor out a negative gcf.
C. List the variables that appear in all (not just some) of the terms and attach the smallest
D.
II.
exponent that appears on that variable.
Use division in order to fill in the ( ).
For example, the GCF of −12𝑥𝑥 3 𝑦𝑦 2 + 16𝑥𝑥 4 + 8𝑥𝑥 2 𝑦𝑦 is −𝟒𝟒𝒙𝒙𝟐𝟐
Therefore, −12𝑥𝑥 3 𝑦𝑦 2 + 16𝑥𝑥 4 + 8𝑥𝑥 2 𝑦𝑦 = −𝟒𝟒𝒙𝒙𝟐𝟐 (3𝑥𝑥𝑦𝑦 2 − 4𝑥𝑥 2 − 2𝑦𝑦)
Count the number of terms to decide which method to use.
A. 4 terms ◊Factor by grouping.
1.
2.
3.
4.
Factor the gcf from the first 2 terms.
Factor the gcf from the third and fourth terms.
Make sure you have the same expression in both sets of ( ).
Factor out the common set of ( ) by writing the set of ( ) that appear twice. Put the remaining
terms in the other set of ( ).
B. 3 terms ◊ Use trial and error (or Reverse FOIL)
1. Determine the signs in the sets of ( ) of the factored form of the trinomial.
a. If the second sign is +, the factored form has 2 of the first sign.
b. If the second sign is -, the factored form has one + and one -.
2. Determine numbers that multiply together to give the first coefficient.
3. Determine numbers that multiply together to give the third coefficient.
4. Arrange these factors so that the Outer + Inner terms combine together to give the middle term
of the trinomial.
Note: If you are factoring correctly, there should be no common factor other than 1 in
the (
). For example, the following cannot appear in the factored form of a polynomial.
(8𝑥𝑥 + 6), (6𝑥𝑥 − 9), (10𝑥𝑥 − 25𝑦𝑦).
C. 2 terms ◊ Use a special rule.
1. 𝑭𝑭𝟐𝟐 − 𝑳𝑳𝟐𝟐 = (𝑭𝑭 − 𝑳𝑳)(𝑭𝑭 + 𝑳𝑳)
(Difference of Squares)
𝟑𝟑
𝟑𝟑
𝟐𝟐
𝟐𝟐
2. 𝑭𝑭 + 𝑳𝑳 = (𝑭𝑭 + 𝑳𝑳)(𝑭𝑭 − 𝑭𝑭𝑭𝑭 + 𝑳𝑳 ) (Sum of Cubes)
3. 𝑭𝑭𝟑𝟑 − 𝑳𝑳𝟑𝟑 = (𝑭𝑭 − 𝑳𝑳)(𝑭𝑭𝟐𝟐 + 𝑭𝑭𝑭𝑭 + 𝑳𝑳𝟐𝟐 ) (Difference of Cubes)
Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 …
Perfect Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 …
III. Make sure no set of ( ) can be factored any further, usually as a difference of squares.