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

##### 1 Answer

Jun 7, 2017

#### Answer:

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

Then

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

What is sufficient is if

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