Question

The number of polynomials $p\left(x\right)$ with integer coefficients such that $p\left(2\right)=3$ and $p\left(6\right)=5$ is

• A. $0$
• B. $2$
• C. $3$
• D. Infinitely many

Answer 1

The correct option is A $0$

Let $p\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0$
$p\left(y\right)=a_n{y}^n+a_{n-1}{y}^{n-1}+\cdots +a_0,y\ne x$

Now, $p\left(x\right)-p\left(y\right)=a_n(x^n-y^n)+a_{n-1}(x^{n-1}-y^{n-1})+\cdots +a_1(x-y)$
Since $x-y$ divides $x^n-y^n\forall n\in N,$
$x-y$ also divides $p\left(x\right)-p\left(y\right).$

$\frac{p\left(6\right)-p\left(2\right)}{6-2}=\frac{2}{4}\notin N$
So no such polynomial exists.