Question

Linear and Quadralic Equations and Inequalities.
Prove that if the value of the quadratic trinomial $a{x}^2-bx+c$ is an integer for $x_{1}0,x_2=1,and{x}_3=2,$ then the value of the given trinomial is an in teger for any integral x.

Answer 1

$f\left(x\right)=a{x}^2-bx+c$

$f\left(0\right)=c\in Z⟹c$ is an integer

$f\left(1\right)=a-b+c\in Z$ and $c\in Z⟹a-b$ is an integer

Hence, $2a-2b$ is an integer

$f\left(2\right)=4a-2b+c$ and $c\in Z⟹4a-2b$ is an integer

Hence, $(4a-2b)-(2a-2b)=2a$ is an integer and hence $a$ is an integer.

And, since $a-b$ is also an integer , hence $b$ is also an integer.

Since, all $a,b$ and $c$ are integers and hence for all integral value of $x$, $f\left(x\right)$ is an integer.