Question

If $x^2+ax+b$ is an integer for every integer $x$ then

• A. $a$ is always an integer but b need not be an integer
• B. $b$ is always an integer but a need not be an integer.
• C. $a+b$ is always an integer.
• D. none of these.

Answer 1

The correct option is C $a+b$ is always an integer.

$f\left(x\right)=x^2+ax+b$ is an integer for every integer x

$\therefore x$ is always integer

For $f\left(x\right)$ to be integer. $x^2+ax+b$ must be integers, for every integer

Let $x^2+ax+b=k,kϵz$

$x^2+ax+(b-k)=0$

Roots of above equation must be integers

$\therefore$Sum of roots also integer

$\therefore -a$ is integer

$\therefore a$ is integer

Product of roots also integer

$\therefore b-k$ is integer

$\therefore b$ is integer ($\because$ k is integer)

$\therefore a+b$ is always an integer.