Polynomial and Prime

Algebra Level 3

Let $f(x)$ be a polynomial of degree $n$ with integer coefficients, where $n$ is a non-negative integer.
Let $m$ be an integer.

True or false:

$f(m)$ is a factor of $f\left( m + k f(m) \right)$ for any integer $k$ .

True only when

k=0

, false otherwise Depends on special value of

n

and

k

and polynomial

f

Always True

2 solutions

İlker Can Erten
Feb 12, 2020

if $f(a)=0$ then the remainder of $f(x+kf(x))$ divided by $f(x)$ is $f(a+kf(a))$ hence $f(a)=0$ the remainder is $0$ which shows that $f(x)$ is a factor of

$f(x+kf(x))$

Muhammad Rasel Parvej
Apr 4, 2018

Not a solution.

The result of this problem proves that a polynomial can not represent only primes.

Polynomial and Prime

2 solutions

0 pending reports