Question

Let P(x) be a non-zero polynomial with integer coefficients. If P(n) is divisible by n for each positive integer n, what is the value of P(0)?
See the answer is 00.
I just want the detailed solution..
10 points.....

Answer 1

P(x) is a polynomial with integer coefficient .
Let , P(x) = axⁿ + a1.x^(n-1) + a2.x^(n-2) ........... + an.1

A/C to question ,
P(n) is divisble by n , it means this is possible only when , if P(x) has no any constant term .
example : P(x) = x³ + x² + x if we put x = n
P(n) = n³ + n² + n = n( n² + n +1) is divisible by n .

hence, P(x) = axⁿ + a1.x^(n-1) + a2.x^(n-2) +......+ a(n-1)x
now you put x = 0
P(0) = a.(0)ⁿ + a1.(0)^(n-1) + a2.(0)^(n-2)+...........+ a(n-1).(0) = 0 + 0 + 0+....+0 = 0

hence, P(0) = 0