Number of polynomials p(x) with integral coefficients satisfying the condition p(1)=2 and p(3)=1 are

Question

$\begin{array}{1 1}(A)\;0&(B)\;1\\(C)\;2&(D)\;\text{infinite}\end{array} $

Answer 1

If $p(x)$ is one such polynomial.

$p_1(x)$ be such that

$p(x)-1=(x-3)p_1(x)$

$\Rightarrow p(1)-1=-2p_1(1)$

$\Rightarrow 1=-2p_1(1)$

$\Rightarrow p_1(1)=-\large\frac{1}{2}$

This is not possible as $p_1(1)$ is an integer.

Hence no such polynomial is possible.

Hence (A) is the correct answer.