Factoring Polynomials with Four Terms
We can rewrite a trinomial as a polynomial with four terms and then used
factoring by grouping. Factoring by grouping can also be used on other types of
polynomials with four terms.
Example 1
Polynomials with four terms
Use grouping to factor each polynomial completely.
a) x^{3} + x^{2} + 4x + 4
b) 3x^{3}  x^{2}  27x + 9
c) ax  bw + bx  aw
Solution
a) Note that the first two terms of x^{3} + x^{2} + 4x
+ 4 have a common factor of x^{2},
and the last two terms have a common factor of 4.
x^{3} + x^{2} + 4x + 4 
= x^{2}(x + 1) + 4(x + 1) 
Factor by grouping. 

= (x^{2} + 4)(x + 1) 
Factor out x + 1. 
Since x^{2} + 4 is a sum of two squares, it is prime and the polynomial is factored
completely.
b) We can factor x^{2} out of the first two terms of 3x^{3} 
x^{2}  27x + 9 and 9 or 9
from the last two terms. We choose 9 to get the factor 3x  1 in each case.
3x^{3}  x^{2}  27x + 9 
= x^{2}(3x  1)  9(3x  1) 
Factor by grouping. 

= (x^{2}  9)(3x  1) 
Factor out 3x  1. 

= (x  3)(x + 3)(3x  1) 
Difference of two squares 
This thirddegree polynomial has three factors.
c) First rearrange the terms so that the first two and the last two have common
factors:
ax  bw + bx  aw 
= ax + bx  aw  bw 
Rearrange the terms. 

= x(a + b)  w(a + b) 
Common factors 

= (x  w)(a + b) 
Factor out a + b. 
