the greatest common factor and factoring by grouping section 6.1
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THE GREATEST COMMON FACTOR AND FACTORING BY GROUPING
Section 6.1
The Greatest Common Factor and Factoring by Grouping
Section 6.1
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
Section 6.1
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 28
2 14
factors
a factored form of x5
a factored form of x5
2 3x x
factorsa factored form of a factored form of
2 3x x
factors
2 5 6x x
Finding the Greatest Common Factor of a List of Integers
Section 6.1
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.
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 45 and 75.
Find the GCF of 36 and 42.
15
45
1 45
3 15
5 9
75
1 75
3 25
5 15
6
Finding the Greatest Common Factor of a List of Integers
Section 6.1
Find the GCF for the expressions1.
2.
The GCF of a list of 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.
4 factors of x in common, or
3 5 and x x
10 and x x
3x
x
0 3
1 2
x x
x x
0 5
1 4
2 3
x x
x x
x x
0 1x x0 10
1 9
2 8
3 7
4 6
5 5
x x
x x
x x
x x
x x
x x
4 7 and x x4
7
x x x x x
x x x x x x x x
4x
Finding the Greatest Common Factor of a List of Integers
Section 6.1
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 numbers1. 72 and 9018
72
8 9
2 4 3 3
2 2
90
10 9
2 53 3
72 2 2 2 3 3
90 2 3 3 5
2 3 3 18
Finding the Greatest Common Factor of a List of Integers
Section 6.1
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 numbers1. 72 and 90
2. 32 and 33
3. 14, 24, and 60
4. 54 and 99
2
1
9
18
Finding the Greatest Common Factor of a List of Terms
Section 6.1
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 monomials1.
2.
3.
6 5 47 and 21m n m n
232 and 40a b abc
235 and 18x y
57m n
8ab
1
Factoring Out the Greatest Common Factor
Section 6.1
The GCF of a polynomial is the GCF of the individual monomial terms.
Find the GCF of 8 14x 2
8 2 2 2
14 2 7
x x
Factoring Out the Greatest Common Factor
Section 6.1
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 the GCF from a polynomial results in a product resembling the distributive property.Factoring the GCF from a polynomial results in a product resembling the distributive property.
Section 6.1
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 (un-
distribute). The given polynomial is
written as a product of the GCF and the result of dividing the polynomial by the GCF.
Factoring Out the Greatest Common Factor
8 14x 2 4 72x
2 4 7x
GCF is 2
8 14 2 4 7x x
Factoring Out the Greatest Common Factor
Section 6.1
Factor the GCF1.
2.
3.
3 215 24k k
2 26 18 12a c abc ac
3 2 2 330 24x yz x y z
23 5 8k k
2 26 5 4x yz xz y
6 3 2ac a b c
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
Section 6.1
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
Section 6.1
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.
22 5 4 10xy y x y
2 02 45 1xy y x y
22 2 55y x y x y
2 525 2x yy x y
2 5 2x y y
Factoring by Grouping
Section 6.1
Factor1.
2.
3.
4 7 28ab a b
2 2 3 2 210 10 15 15a b b a b b
7 4a b
25 2 3b b a b
3 22 2 3x x x cannot be factored