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 #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.