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

To prove that the polynomial \( f(x) = 0 \) cannot have any integral root given that \( f(1) \) and \( f(2) \) are both odd integers, we can follow these steps:

### Step 1: Assume \( f(x) \) has an integral root
Let us assume that \( f(x) = 0 \) has an integral root \( k \). This means that \( f(k) = 0 \).

### Step 2: Factor the polynomial
Since \( k \) is a root, we can express the polynomial \( f(x) \) as:
\[
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