Factoring

1
Factoring Using the Distributive Property
2
Distributive Property
Apply the distributive property to 3c(4c 2).
In this example, the result of the
multiplication, 12c2 6c, is the product.
The factors in this problem were the monomial,
3c, and the polynomial, (4c-2).
3c(4c-2) is the factored form of 12c2 6c.
Polynomials can be written in factored form by
reversing the process of the distributive
property.
3
Factoring
Factor the following polynomial using the
distributive property.
Step 1 Find the GCF for both terms.
9m3n2 3 ? 3 ? m ? m ? m ? n ? n
24mn4 2 ? 2 ? 2 ? 3 ? m ? n ? n ? n ? n
The GCF is 3mn2.
4
Factoring
Step 2 Divide each term of the polynomial by
the GCF.
5
Factoring
Step 3 Write the polynomial as the product of
the GCF and the remaining factor of each term
using the distributive property.
6
Factoring Polynomials
What happens when the distributive property is
applied to a problem such as (2a 3b)(5c 8d)?
This problem can be rewritten by distributing
each term in the first set of parentheses with
each term in the second set of parentheses as
follows.
We now have a polynomial with four terms and
there doesnt appear to be a factor that is
common to all four terms.
7
Factoring Polynomials
How can we factor 4ab 2ac 8xb 4xc?
The answer is by grouping terms which do have
something in common. Often, this can be done in
more than one way. For example
or
Next, find the greatest common factor for the
polynomial in each set of parentheses.
The GCF for (4ab2ac) is 2a. The GCF for (8xb
4xc) is 4x.
The GCF for (4ab 8xb) is 4b. The GCF for (2ac
4xc) is 2c.
8
Factoring Polynomials
Divide each polynomial in parentheses by the GCF.
9
Factoring Polynomials
Write each of the polynomials in parentheses as
the product of the GCF and the remaining
polynomial.
Apply the distributive property to any common
factors.
Factor further if necessary.
Notice that it did not matter how the terms were
originally grouped, the factored forms of the
polynomials are identical.
10
You Try It
1. Find the GCF of the terms in each expression.
a. 4x2 6xy b. 60a2 30ab 90ac
2. Factor each polynomial.
a. 6tt 42ts
b. 7a2 4ab 12b4 21ab3
11
Problem 1

• 4x2 2 ? 2 ? x ? x
• 6xy 2 ? 3 ? x ? y

The GCF is 2x .

• 60a2 2 ? 2 ? 3 ? 5 ? a ? a
• 30ab 2 ? 3 ? 5 ? a ? b
• 90ac 2 ? 3 ? 3 ? 5 ? a ?
  c

The GCF is 30a .
12
Problem 2

• 6t2 2 ? 3 ? t ? t
• 42ts 2 ? 3 ? 7 ? t ? s

The GCF is 6t.
Divide each term by the GCF.
Write the polynomial as the product of the GCF
and the remaining factor of each term using the
distributive property.
13
Problem 2
Use grouping symbols to group terms which have a
common factor.
Divide the polynomial in each set of parentheses
by its common factor.
Factor out a (-1) from the second set of
parentheses.
Apply the distributive property.