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(D) Let $f(x) = x^2 + ax + b$. Since $f(x)$ is an integer for every integer $x$,we have: $f(0) = 0^2 + a(0) + b = b$. Since $f(0)$ must be an integer,

Vedclass · Accepted Answer

None of these.

Vedclass · Answer

(D) Let $f(x) = x^2 + ax + b$. Since $f(x)$ is an integer for every integer $x$,we have: $f(0) = 0^2 + a(0) + b = b$. Since $f(0)$ must be an integer,$b$ must be an integer. Now,$f(1) = 1^2 + a(1) + b = 1 + a + b$. Since $f(1)$ is an integer and $b$ is an integer,$1 + a + b$ is an integer,which implies $a$ must be an integer. Therefore,both $a$ and $b$ must be integers. Since the options provided do not state that both are integers,the correct choice is $D$.

If $x^2 + ax + b$ is an integer for every integer $x$,then which of the following is true?

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