# Factor Polynomials by Common Factor Questions With Solutions

How to factor a polynomial using a common factor? Questions are presented along with detailed solutions and explanations are included.

## What is factorization by common factor?

It is a factorization method based on the law of distributivity which is
a(b + c) = a � b + a � c

and used in
reverse as follows
a � b + a � c = a(b + c)

a is a common factor to a b and a c is therefore factored out.
Factoring a polynomial is to write it as the product of simpler polynomials.
Example:
2
x + 4 = 2(x + 2)
3
x 2 - x = x(3x - 1)
NOTE: it is very easy to check if your factorization is correct by expanding the factored form to check if you get the original polynomial
Example: check that 3
x 2 - x = x(3x - 1)
Expand
x(3x - 1) using the law of distributivity.
x(3x - 1) = (x)(3x) +(x)(-1) = 3x2 - x , which is correct.

## More Examples

Find a common factor and use the method of distributivity in reverse to factor the polynomials completely.
a)
9 x - 6
b)
x 2 - x
c)
3 x + 12 x y
d)
16 x 3 + 8 x 2 y + 4 x y 2
e)
2 x 4(x + 5) + x 2(x + 5)

## Solution to the above examples

a) Find any common factors in the two terms of
9 x - 6 by expressing both terms 9x and 6 in the given binomial as prime factorization. Hence
9 x - 6 = 3 �3 �x - 2 �3
The greatest common factor is
3 and is factored out. Hence
9 x - 6 = 3 (3 x - 2)

b) The prime factorization of
x 2 and x is needed to find the greatest common factor in x 2 - x.
x 2 - x = x � x - x = x � x - 1 � x
The greatest common factor is
x and is therefore factored out. Hence
x 2 - x = = x (x - 1)

c) The prime factorizations of
3 x and 12 x y are needed to find the greatest common factor in 3 x + 12 x y.
3 x + 12 x y = 3 � x - 3 � 4 � x � y = 3 � x � 1 - 3 x � 4 � y
The greatest common factor is
3 x. Hence
3 x + 12 x y = 3 x (1 + 4 y)

d) The prime factorization of
16 x 3 , 8 x 2 y and 4 x y 2 are needed to find the greatest common factor in 16 x 3 + 8 x 2 y + 4 x y 2.
16 x 3 + 8 x 2 y + 4 x y 2 = 2 � 2 � 2 � 2 � x � x � x + 2 � 2 � 2 � x � x � y + 2 � 2 � x � y � y
The greatest common factor is
2 � 2 � x = 4 x. Hence
16 x 3 + 8 x 2 y + 4 x y 2 = 4 x ( 2 � 2 � x � x + 2 � x � y + y � y) = 4 x (4 x 2 + 2 x y + y 2)

e) We note that
x + 5 is a common factor which can be factored out as follows:
2 x 4(x + 5) + x 2(x + 5) = (x + 5)(2 x 4 + x 2)
We now find the greatest common factor of the terms
2 x 4 and x 2 and factor it out.
2 x 4 + x 2 = 2 � x � x � x � x + x � x = x 2(2 x 2 + 1)
The complete factoring of
2 x 4(x + 5) + x 2(x + 5) is written as follows:
2 x 4(x + 5) + x 2(x + 5) = x 2(x + 5)(2 x 2 + 1)

## Questions

Use common factors to factor completely the following polynomials
a)
- 3 x + 9
b)
28 x + 2 x 2
c)
11 x y + 55 x 2 y
d)
20 x y + 35 x 2 y - 15 x y 2
e)
5 y (x + 1) + 10 y 2(x + 1) - 15 x y (x + 1)

### Solutions to the Above Questions

a) Find any common factors in the two terms of - 3 x + 9 by expressing both terms 3 x and 9 in the given binomial as prime factorization.
- 3 x + 9 = - 3 � x - 3 � 3
The greatest common factor is 3 and is factored out. Hence
- 3x + 9 = 3 (- x + 3) = - 3 (x - 3)

b) Write the prime factorization of each of the terms in the given polynomial 28 x + 2 x 2.
28 x + 2 x 2 = 2 � 2 � 7 � x + 2 � x � x
The greatest common factor is 2 x and is factored out. Hence
28 x + 2 x 2 = 2 x (14 + x)

c) Write the prime factorization of each of the terms in the given polynomial 11 x y + 55 x 2 y.
11 x y + 55 x 2 y = 11 � x � y + 5 � 11 � x � x � y
The greatest common factor is 11 x y and is factored out. Hence
11 x y + 55 x 2 y = 11 x y(1 + 5 x)

d) Write the prime factorization of each of the terms in the given polynomial 20 x y + 35 x 2 y - 15 x y 2.
20 x y + 35 x 2 y - 15 x y 2 = 2 � 2 � 5xy + 5 � 7 � x � x � y - 3 � 5xy � y
The greatest common factor is 5 x y and is factored out. Hence
20 x y + 35 x 2 y - 15 x y 2 = 5 x y( 4 + 7 x - 3 y)

e) We start by factoring out the common factor (x + 1) in the given polynomial.
5 y (x + 1) + 10 y 2(x + 1) - 15 x y (x + 1) = (x + 1)(5y + 10y2 - 15 x y)
We now factor the polynomial 5y + 10y2 - 15 x y using the GCF to all three terms.
5 y + 10y2 - 15 x y = 5 � y + 2 � 5 � y � y - 3 � 5 � y � x = 5 � y (1 + 2 y - 3 x)
The given polynomial may be factored as follows.
5 y (x + 1) + 10 y 2(x + 1) - 15 x y (x + 1) = 5 y(x + 1)(1 + 2y - 3 x)