STD 11 - 4.2 Quadratic equations and inequations — Maths STD 11 Science — Question

Let

$P(x)=a_n x^n+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\ldots +a_1 x+a_0$

$a_0, a_1, a_2 \ldots \in I$

Given, $P(2)=2$ and $P(4)=5$

$2=a_n 2^n+a_{n-1} 2^{n-1}+a_{n-2} 2^{n-2}+\ldots +a_1 2+a_0 \ldots (i)$

$5=a_n 4^n+a_{n-1} 4^{n-1}+a_{n-2} 4^{n-2}+\ldots +4 a_1+a_0 \ldots (ii)$

On subtracting Eq.$(i)$ from Eq.$(ii)$, we get $3=a_n\left(4^n-2^n\right)+a_{n-1}\left(4^{n-1}-2^{n-1}\right) +4 a_1+a_0 \ldots (ii)$

Clearly,$LHS$ is odd number and $RHS$ is even number.

$\therefore$ No polynomials exists.