Which of the following is a correct factorization of x^4 - x^3 - 7x^2 + x + 6?

• A. (x - 1)(x + 1)(x - 3)(x + 2)
• B. (x - 1)(x + 1)(x - 2)(x + 3)
• C. (x - 2)(x + 2)(x - 3)(x + 1)
• D. (x + 1)(x + 1)(x - 3)(x - 2)

Answer: (A)

Start with the idea that factoring a polynomial often means finding its roots and turning those roots into linear factors. For a quartic with integer constant term, test possible rational roots: ±1, ±2, ±3, ±6.

First test x = 1: plug in to x^4 - x^3 - 7x^2 + x + 6 and you get 1 - 1 - 7 + 1 + 6 = 0, so x - 1 is a factor. Divide to get a cubic: x^3 - 7x - 6.

Now test a root of the cubic. Try x = -2: (-8) + 14 - 6 = 0, so x + 2 is a factor. Divide again to obtain x^2 - 2x - 3, which factors as (x - 3)(x + 1).

Putting it all together gives (x - 1)(x + 2)(x - 3)(x + 1). Rearranging the factors, that matches the given factorization (x - 1)(x + 1)(x - 3)(x + 2). Verifying by multiplying the pairs confirms it expands back to x^4 - x^3 - 7x^2 + x + 6.