Question

What are necessary and sufficient conditions for a linear polynomial to be a factor of a given polynomial?

Answer 1

See explanation...

Explanation:

Suppose you are given a polynomial:

`f(x) = a_n x^n + a_(n-1) x^(n-1) +...+ a_1 x + a_0`

and a potential linear factor:

`px+q`

where all of `a_n`, `a_(n-1)`,..., `a_0`, `p` and `q` are integers.

Then `px+q` can only be a factor of `f(x)` if `p` is a factor of `a_n` and `q` is a factor of `a_0`.

This immediately allows you to rule out many possibilities, but is not a sufficient condition.

What is sufficient is if `f(-q/p) = 0`.

In fact this last condition is both necessary and sufficient regardless of whether the coefficients are integers, rational, real or even complex numbers.