Finding the Greatest Common Factor of Polynomials
In a multiplication problem, the numbers multiplied together are called factors. The answer to a
multiplication problem is called the product.
, 5 and 4 are factors and 20 is the product.
In the multiplication problem
, we say we have factored 20 into
If we reverse the problem,
.
In this worksheet we will factor polynomials.
In the multiplication problem
(
,
)
If we reverse the problem,
(
are the factors and
) , we say we have factored
is the product.
into
.
Name the factors and the product in each problem.
1.
(
)
factors: __________________________
product: __________________
2.
(
)
factors: __________________________
product: __________________
factors: __________________________
product: __________________
factors: __________________________
product: __________________
3.
(
4.
(
)
)
The first step in factoring polynomials is to factor out the greatest common
factor (GCF). This is the largest integer and highest degree of each variable
that will divide evenly into each term of the polynomial.
Factoring is the reverse of multiplying!

In the polynomial
, 5 is the largest integer that will divide 5x and 35, and we cannot factor out
any variable because the second term, 35, does not have a variable part.
To factor

).
(
In the polynomial
, 3 is the largest integer that will divide
. We can factor out
because each term has at least one factor of (look for the term with the lowest degree of each
variable).
To factor

we write:
we write:
(
).
In the polynomial
, 4 is the largest integer that will divide
out and
because each term has at least one factor of and two factors of .
To factor
we write:
(
. We can factor
).
___________________________________________________________________________________________________________
Finding the Greatest Common Factor of Polynomials
Find the largest integer that will divide all the terms.
6.
Find the largest degree of
7.
8.
that can be factored out of all the terms.
10.
11.
12.
Factor the polynomials.
13.
=
14.
15.
16.
To factor polynomials, find the greatest common factor (GCF) of the coefficients and factor it out- divide
each term by the GCF. Then find the greatest common factor (GCF) of the variables by finding the lowest
power of each variable that will divide all terms and factor it out- divide each term by GCF. Move the GCF
to the outside and write in parenthesis what is remaining, after you factor out the GCF.
Factor each of the following polynomials.
17.
18.
19.
20.
21.
22.
23.
24.
25.
If the leading coefficient is negative, always factor out the negative!
26.
27.
28.
Finding the Greatest Common Factor of Polynomials
In a multiplication problem, the numbers multiplied together are called factors. The answer to a
multiplication problem is called the product.
, 5 and 4 are factors and 20 is the product.
In the multiplication problem
, we say we have factored 20 into
If we reverse the problem,
.
In this worksheet we will factor polynomials.
In the multiplication problem
(
If we reverse the problem,
are the factors and
,
)
(
) , we say we have factored
is the product.
into
.
Name the factors and the product in each problem.
5x-35
5, (x-7)
1.
(
)
factors: __________________________
product: __________________
2.
(
)
3x, (x+9)
factors: __________________________
3x^2+27x
product: __________________
-10x, (x-6)
factors: __________________________
-10x^2+60x
product: __________________
4xy^2, (3x+8y)
factors: __________________________
12x^2y^2+32xy^3
product: __________________
3.
(
4.
(
)
)
The first step in factoring polynomials is to factor out the greatest common
factor (GCF). This is the largest integer and highest degree of each variable
that will divide evenly into each term of the polynomial.
Factoring is the reverse of multiplying!

In the polynomial
, 5 is the largest integer that will divide 5x and 35, and we cannot factor out
any variable because the second term, 35, does not have a variable part.
To factor

).
(
In the polynomial
, 3 is the largest integer that will divide
. We can factor out
because each term has at least one factor of (look for the term with the lowest degree of each
variable).
To factor

we write:
we write:
(
).
In the polynomial
, 4 is the largest integer that will divide
out and
because each term has at least one factor of and two factors of .
To factor
we write:
(
. We can factor
).
___________________________________________________________________________________________________________
Finding the Greatest Common Factor of Polynomials
Find the largest integer that will divide all the terms.
6.
7.
Find the largest degree of
5
6
7
09
8.
that can be factored out of all the terms.
10.
0
11.
x
12.
x^3
x
Factor the polynomials.
13.
=
14.
9(x+5)
7x(x-3)
15.
16.
6x^3(3x^3+2)
5x(3x^2-5x+11)
To factor polynomials, find the greatest common factor (GCF) of the coefficients and factor it out- divide
each term by the GCF. Then find the greatest common factor (GCF) of the variables by finding the lowest
power of each variable that will divide all terms and factor it out- divide each term by GCF. Move the GCF
to the outside and write in parenthesis what is remaining, after you factor out the GCF.
Factor each of the following polynomials.
17.
18.
6x(x-4)
20.
19.
7x(2x-5)
x(5x+1)
21.
4x(5x+11)
23.
22.
9x(4x^2+7x-3)
17x(x+3)
24.
3x^3y^2(x+5y)
25.
5y^2(4y^2-3y+6)
3x^2y^5(3x^5-y)
If the leading coefficient is negative, always factor out the negative!
26.
27.
-2m(m^3-7m+3)
28.
-5xy(x-7)
-(x^2-5x+6)