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# 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.