1. The Greatest Common Factor and Factoring by Grouping
Section 6.1
The Greatest Common Factor and Factoring by Grouping

2. 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.
Section 6.1

3. 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
factors
a factored form of x5
factors
a factored form of
factors
Section 6.1

4. Finding the Greatest Common Factor of a List of Integers
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. 45 1 3 15 5 9 75 1 3 25 5 15
Section 6.1

5. Finding the Greatest Common Factor of a List of Integers
Find the GCF for the expressions
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
Section 6.1

6. 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
72 and 90 72 90 8 9 10 9 2 4 3 3 3 3 2 5 2 2
Section 6.1

7. 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
72 and 90
32 and 33
14, 24, and 60
54 and 99
Section 6.1

8. 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
Section 6.1

9. Factoring Out the Greatest Common Factor
The GCF of a polynomial is the GCF of the individual monomial terms.
Find the GCF of
Section 6.1

10. 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!
Section 6.1

11. 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 (un-distribute).
The given polynomial is written as a product of the GCF and the result of dividing the polynomial by the GCF.
GCF is 2
Section 6.1

12. Factoring Out the Greatest Common Factor
Factor the GCF
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.
Section 6.1

13. 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.
Section 6.1

14. 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.
Section 6.1

15. Factoring by Grouping
Factor
Section 6.1