Question

If $x^2+px+q$ is an integer for all integral values of $x$, then

• A. $p$ is an integer
• B. $q$ is an integer
• C. $\frac{p+q}{2}$ is an integer
• D. $\frac{p-q}{2}$ is an integer

Answer 1

Answer: (A), (B)

The correct options are
A $p$ is an integer
B $q$ is an integer

The equation will have integral values for every integral value of $x$ if,;

$x=k$, $k\in Z$

Substituting $x=k$ in the equation :

$\Rightarrow k^2+pk+q$, this equation will be integer if and only if $p$ and $q$ both are integers

$\therefore p,q\in Z$

$\frac{(p+q)}{2}$ can be integer only if $p+q$ is a multiple of $2$.

Here $p,q$ are only integers and there sum can or cannot be a multiple of $2$ always.

Similarly option D is wrong.