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) x3 + x2 + 4x + 4
b) 3x3 - x2 - 27x + 9
c) ax - bw + bx - aw
Solution
a) Note that the first two terms of x3 + x2 + 4x
+ 4 have a common factor of x2,
and the last two terms have a common factor of 4.
x3 + x2 + 4x + 4 |
= x2(x + 1) + 4(x + 1) |
Factor by grouping. |
|
= (x2 + 4)(x + 1) |
Factor out x + 1. |
Since x2 + 4 is a sum of two squares, it is prime and the polynomial is factored
completely.
b) We can factor x2 out of the first two terms of 3x3 -
x2 - 27x + 9 and 9 or -9
from the last two terms. We choose -9 to get the factor 3x - 1 in each case.
3x3 - x2 - 27x + 9 |
= x2(3x - 1) - 9(3x - 1) |
Factor by grouping. |
|
= (x2 - 9)(3x - 1) |
Factor out 3x - 1. |
|
= (x - 3)(x + 3)(3x - 1) |
Difference of two squares |
This third-degree 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. |
|