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.

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 · 2 = 2 (x + 2) the common factor is 2
3 x^{ 2} - x = 3 x · x - 1 · x = x (3x - 1) the common factor is x

NOTE: that 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) by expand x (3x - 1) using the law of distributivity as follows:
x (3x - 1) = (x) (3x) +(x) (-1) = 3 x^{2} - 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 · 5 · x · y + 5 · 7 · x · x · y - 3 · 5 · x · y · 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 + 10y^{2} - 15 x y)
We now factor the polynomial 5y + 10y^{2} - 15 x y using the GCF to all three terms.
5 y + 10y^{2} - 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)