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Integer Roots (Tomato Subjective 175)

Problem:- Let \(\text{P(x)}=x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_{1}x+a_{0}\) be a polynomial with integer coefficients,such that,\(\text{P(0) and P(1)}\)  are odd integers.Show that

(a)\(\text{P(x)}\) does not have any even integer roots. 

(b)\(\text{P(x)}\) does not have any odd integer roots.

Solution:- Given the two statements (a) and (b)  above it is clear that if we can prove that \(\text{P(x)}\) has no integer roots,then we are done.

Proof:- Let us assume \(\text{P(x)}\) has an integer root \(\text{ a}\).

Then we can write,

$$ \text{P(x)=(x-a)Q(x)}\dots(1)$$

where,\(\text{Q}\)is any function of \(\text{ x}\).


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