Greatest Common Factor and
Factoring by Grouping
6.1
Find the Greatest Common
Factor (GCF) of a Set of Terms
Factored form: A number or an expression
written as a product of factors.
Examples:
2x + 8 = 2(x + 4)
x2 + 5x + 6 = (x + 2)(x + 3)
Greatest common factor (GCF) of a set of
terms: A monomial with the greatest
coefficient and degree that evenly divides
all of the given terms.
Method
To find the GCF of two or more monomials,
1. Write the prime factorization in exponential form
for each monomial.
2. Write the GCF’s factorization by including the
prime factors (and variables) common to all the
factorizations, each raised to its smallest exponent
in the factorizations.
3. Multiply the factors in the factorization created in
step 2.
Note: If there are no common prime factors, then
the GCF is 1.
Exercise 1
Find the GCF
[06] 6m4n9,
[12] 8(m – n),
15mn5
11(m – n),
m–n
A Method to Find the Prime
Factorization of a Constant
Determine how many times two can go into
the constant, then the number of threes,
fives, sevens, elevens, etc.
Example: Factor 420
2 ) 420
2 ) 210
3 ) 105
5 ) 35
7) 7
1
There are 2 twos, 1 three, 1 five, 1 seven
420 = 22315171 = 22 3 5 7
Factor a Monomial GCF Out of
the Terms of a Polynomial
The method to factor a monomial GCF out of
the terms of a polynomial is:
1. Find the GCF of all the terms in the
polynomial.
2. Rewrite the polynomial as a product of the
GCF and the quotient of the polynomial
and the GCF.
Exercise 2
Factor:
[20] -2x2y2 - 8x3y + 2xy
[30] a( b – 5 ) +
7(b–5)
Factor Polynomials by Grouping
Method to factor a four-term polynomial by grouping:
2x2 - 4x - 4x + 8
1. Factor out any monomial GCF (other than 1) that is
common to all four terms.
2. Group together pairs of terms and factor the GCF
out of each pair.
3. If there is a common binomial factor, then factor it
out.
4. If there is no common binomial factor, then
interchange the middle two terms and repeat the
process. If there is still no common binomial factor,
then the polynomial cannot be factored by
grouping.
Exercise 3
Factor:
[50] 15 c3 – 5 c2d + 6 cd2 – 2 d3
[62] 12 ac + 12 cx - 3 ac2 - 3 c2x
[**] 2x3 – y3 – 2x2y + xy2
Greatest Common Factor and
Factoring by Grouping
6.1