1. Tardigrade
  2. Question
  3. Mathematics
  4. 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

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Q. $P ( x )$ 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

Complex Numbers and Quadratic Equations

Solution:

$ 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.