Let-f-x-be-a-quadratic-polynomial-with-integer-coefficients-such-that-f-0-and-f-1-are-odd-integers-Prove-that-the-equation-f-x-0-does-not-have-an-integer-solution-

Question Number 19245 by Tinkutara last updated on 07/Aug/17

Let f (x) be a quadratic polynomial

with integer coefficients such that f (0)

and f (1) are odd integers . Prove that

the equation f (x) = 0 does not have an

integer solution .

Commented by RasheedSindhi last updated on 08/Aug/17

Let f (x) = {ax}^{2} + bx + c

f (0) = c (odd)

f (1) = a + b + c (odd)

c \in O \land a + b + c \in O \Rightarrow a + b \in E

a + b \in E \Rightarrow {\begin{cases} a, b \in E \\ a, b \in O \end{cases}

Case - 1 : a, b \in E \land c \in O

Case - 2 : a, b, c \in O

Continue

Answered by RasheedSindhi last updated on 09/Aug/17

Quadratic polynomial whose roots

are p and q is

f (x) = x^{2} - (p + q) x + pq

f (0) = pq \in O (given) \Rightarrow p, q \in O

f (1) = 1 - (p + q) + pq \in O (given)

But

p, q \in O \Rightarrow p + q \in E and

f (1) = 1 - (p + q) + pq \in O

\Rightarrow (odd) - (even) + (odd)

\Rightarrow (odd) + (odd) = (even) \notin O

This contradiction proves

that p and q are neither odd

nor even or they are not integers .

Commented by Tinkutara last updated on 09/Aug/17

Thank you very much Sir!