Let f(x) be a polynomial with integral coefficients. If f(1) and f(2) both are odd integers, prove that f(x) = 0 can' t have any integral root.

Explanation

Let us assume that f(x) = 0 for some integer x = k
Then, x - k divides f(x).
`therefore f(x) = (x-k) g (x),` where g(x) is a polynomial with integral coefficients
`rArr` f(1) = (1 - k) g(1) and f(2) = (2-k) g(2)
`rArr ` f(1) f(2) = (1-k)(2-k)g (1)g(2)
which is clearly an even number, which contradicts the given information that both f(1) and f(2) are odd integers . Hence f(x) = 0 con't have any integral root.