Question

If P(x) is a polynomial with integer coefficients such that for 4 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

Correct Answer - 4
`P(a) = P(b) = P(c) = P(d) = 3`
`rArr P(x) = 3` has a, b, c, d as its roots
`rArr 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) re 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.