Next year polynomial

Algebra Level 3

$P(x)$ is a 2018-degree monic polynomial such that $P(x)=2018$ for $x=1,2,3,...,2017$ and $P(2018)=0$ . What is the value of $\large P\left(2018+\frac{2018}{2017!}\right)?$

The answer is 2018.

1 solution

Akeyla Naufal
Dec 14, 2017

Let $Q(x)=P(x)-2018$ , then $Q(x)$ is a 2018-degree monic polynomial with $1,2,3,...,2017$ as its roots.
So , $Q(x)=(x-1)(x-2)(x-3)...(x-2017)(x-k)$ , for a constant $k$ .
Substitute $x=2018$ , we get $-2018=2017.2016.2015.....1.(2018-k)$ , where we get $k=2018+\frac{2018}{2017!}$
Therefore , $P(2018+\frac{2018}{2017!})=Q(2018+\frac{2018}{2017!})+2018=0+2018=2018$

Next year polynomial

The answer is 2018.

1 solution

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