Transcript Factoring GCF and Grouping

Section 14.1
THE GREATEST COMMON FACTOR AND
FACTORING BY GROUPING
The Greatest Common Factor and Factoring by Grouping
 Find the greatest common factor of a list of integers.
 Find the greatest common factor of a list of terms.
 Factor out the greatest common factor from a
polynomial.
 Factor a polynomial by grouping.
The Greatest Common Factor and Factoring by Grouping
 Factored Form
 A number or expression is said to be factored when written as
a product of factors.
a factored form of 28
a factored form of x5
2 14
x 2 x3
factors
factors
a factored form of x  5 x  6
2
 x  2 x  3
factors
Finding the Greatest Common Factor of a List of Integers
 Greatest Common Factor

When given a set of two or more
numbers, the largest natural
number that evenly divides all
the numbers in the set is called
the greatest common factor.
 Find the GCF of 45 and 75.
45
15
75
1
45
1
75
3
15
3
25
5
9
5
15
 To find the GCF using factor
pairs:


list all factor pairs for each
number
select the largest number that
appears in both lists
 Find the GCF of 36 and 42.
6
Finding the Greatest Common Factor of a List of Integers
 Find the GCF for the
expressions
1.
x0
x1
2.
x
0
x 3 and x5
x3 x 0
x 2 x1
x2
x5
x4
x and x
x
common variables raised
to powers is the smallest
exponent in the list.

We can extend this idea by
using what is known as
the prime factorization.

x 4 and x 7
x3
10
1
x3
 The GCF of a list of
x0
x1
x2
x10
x9
x8
x3
x4
x7
x6
x5
x5
x
x4  x x x x
x7  x x x x x x x

4 factors of x in common,
or x 4
Finding the Greatest Common Factor of a List of Integers
 To find the GCF using
 Find the GCF for the
prime factorization:



Find the prime
factorization of each
number using a factor
tree.
Determine which factors
the numbers have in
common.
The GCF will be the
product of each common
prime factor.
numbers
72 and 90 18
1.
72
90
8
2
9
4
3
3
3
2
2
9
10
5
2
72  2 2 2 3 3
90  2
335
2 3 3  18
3
Finding the Greatest Common Factor of a List of Integers
 To find the GCF using
prime factorization:



Find the prime
factorization of each
number using a factor
tree.
Determine which factors
the numbers have in
common.
The GCF will be the
product of each common
prime factor.
 Find the GCF for the
numbers
1.
72 and 90 18
2.
32 and 33
3.
14, 24, and 60
1
2
4.
54 and 99
9
Finding the Greatest Common Factor of a List of Terms
 In general, the GCF of a
list of monomials, is the
product of the GCF for
the coefficients and the
variables.
 Find the GCF of the
monomials
1.
7m6 n and 21m5 n 4
7m 5 n
2. 32a
2
b and 40abc
8ab
3. 35 x
2
and 18 y
1
Factoring Out the Greatest Common Factor
 The GCF of a polynomial is the GCF of the individual
monomial terms.
 Find the GCF of
8x 14
2
 8x  2 2 2 x 


14

2
7


Factoring Out the Greatest Common Factor
 Factored Form
 A number or expression is said to be factored when written as
a product of factors.
 Factoring is answering, “what can I multiply to get
the given expression?”

Your answer will look like a multiplication problem like the
ones from Chapter 5!
Factoring Out the Greatest Common Factor
Factoring the GCF from a polynomial results in a product
resembling the distributive property.
 To factor a monomial GCF
out of a given polynomial



Find the GCF of the terms in
the polynomial.
Write the terms as a product
containing the GCF.
Factor out the GCF (undistribute).

The given polynomial is written
as a product of the GCF and the
result of dividing the
polynomial by the GCF.
GCF
is 2
8x 14
2 4x  2 7
2  4x  7
8x  14  2  4 x  7 
Factoring Out the Greatest Common Factor
 Factor the GCF
1.
15k 3  24k 2
3k 2  5k  8
2.
30 x3 yz 2  24 x 2 y 3 z

6 x 2 yz 5 xz  4 y 2
3.
6a 2c  18abc  12ac 2

6ac  a  3b  2c 
A GCF of 6ac is fine, but we
really don’t like to see (-a…
If the first term is negative,
it is best to take out a
negative GCF, even if it is
just -1.
Factoring by Grouping
 Factoring the GCF is only one stage of factoring.
Sometimes a polynomial can be factored further.
 Polynomials with four terms are factored with a
process called grouping.
Factoring by Grouping
 To factor by grouping




Factor the GCF from all
terms if possible
Group the terms into pairs
Factor the GCF from each
pair
Factor out the common
binomial factor from each
group.

If the remaining binomials
are not common:



Try rearranging the terms
before grouping.
You did not remove the
correct GCF.
The polynomial cannot be
factored.
2 xy  5 y  4 x  10 y
2
2 xy  5 y  4 x  10 y
2
y  2 x  5 y   2  2 x  5 y 
y  2 x  5 y   2  2 x  5 y 
 2 x  5 y  y  2
Factoring by Grouping
 Factor
1.
ab  4a  7b  28
 a  7 b  4
2. 10a
b  10b3  15a 2b  15b 2
2 2

5b  2b  3 a 2  b
3. 2 x  2 

x3  3x 2
cannot be factored