Question

Let $f(x)=A x^2+B x+C$, where $A, B, C$ are real numbers. Prove that if $f(x)$ is an integer whenever $x$ is integer, then the numbers $2 A, A+B$ and $C$ are all integers. Conversely, prove that if the numbers $2 A, A+$ $B$ and $C$ are all integer then $f(x)$ is an integer whenever $x$ is an integer.

Answer 1

SOLUTION : (i)$f(x)=A x^2+B x+C  \Rightarrow  x \in I \text { and } f(x) \in I \\$

$\text { at } x=0, f(0)=C  \Rightarrow C \text { is integer at } x=1, f(1)=A+B+C \\$

$\because \quad C \text { is integer }  \therefore  A+B \text { is also integer } \\$

$\text { at } x=-1, f(-1)=A-B+C  \Rightarrow  f(1)+f(-1)=2 A+2 C \\$

$\because \quad C \quad \text { is integer }  \therefore  2 A \text { is also integer }$

(ii) $f(x)=A x(x-1)+(A+B) x+C \Rightarrow f(x)=2 A \frac{x(x-1)}{2}+(A+B) x+C$

If $x$ is an integer then $\frac{x(x-1)}{2}$ is also an integer and $2 A,(A+B), C \in I$ $\Rightarrow \quad f(x)$ is also an integer.