jmc

Question

Let Q be a polynomial Q(x)=a_0+a_1x++a_nx^n,where a_0,,a_n are nonnegative integers. Given that Q(1)=4 and Q(5)=152, find Q(6).

Accepted Answer

If n 4, then Q(5) 5^4 = 625, so n 3, and we can write Q(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0.We have that Q(1) = a_3 + a_2 + a_1 + a_0 = 4, so a_i 4 for all i. Also, Q(5) = 125a_3 + 25a_2 + 5a_1 + a_0 = 152.Clearly, a_3 1. If a_3 = 0, then 25a_2 + 5a_1 + a_0 = 152. But 25a_2 + 5a_1 + a_0 25 4 + 5 4 + 4 = 125, so a_3 = 1. Then 25a_2 + 5a_1 + a_0 = 27.Clearly, a_2 1. If a_2 = 0, then 5a_1 + a_0 = 27. But 5a_1 + a_0 5 4 + 4 = 24, so a_2 = 1. Then 5a_1 + a_0 = 2.It follows that a_1 = 0 and a_0 = 2, so Q(x) = x^3 + x^2 + 2.In particular, Q(6) = 6^3 + 6^2 + 2 = 254.