# Solutions to Factoring Polynomials by Common Factor Questions With Detailed Solutions

Solutions on how factor polynomials by common factor are presented.

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)

Solution
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 + 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)