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 + 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)