Question

$P(x)$ is a polynomial with integral coefficients such that for four distinct integers $a,b,c,d,P(a)=P(b)=$ $P(c)=P(d)=3$. If $P(e)=5$ (e is an integer), then

• A. $e=1$
• B. $e=3$
• C. $e=4$
• D. No real value of e

Answer 1

Answer: (D)

$P(a)=P(b)=P(c)=P(d)=3$
$\Rightarrow P(x)=3$ has $a,b,c,d$ as its roots
$\Rightarrow P(x)-3=(x-a)(x-b)(x-c)(x-d)Q(x)$
$[\because Q(x)$ has integral coefficient]
Given $P(e)=5$, then
$(e-a)(e-b)(e-c)(e-d)Q(e)=5$
This is possible only when at least three of the five integers $(e-a),(e-b)(e-c),(e-d),Q(e)$ are equal to 1 or $-1$. Hence, two of them will be equal, which is not possible.
Since $a,b,c,d$ are distinct integers. $P(e)=5$ is not possible.