MathDB

n

is a positive integer,

K

the set of polynoms of real variables

x_1,x_2,...,x_{n+1}

and

y_1,y_2,...,y_{n+1}

, function

f:K\rightarrow K

satisfies f(p+q)=f(p)+f(q), f(pq)=f(p)q+pf(q), (\forall)p,q\in K. If f(x_i)=(n-1)x_i+y_i, f(y_i)=2ny_i for all

i=1,2,...,n+1

and

\prod_{i=1}^{n+1}(tx_i+y_i)=\sum_{i=0}^{n+1}p_it^{n+1-i}

for any real

t

, prove, that for all

k=1,...,n+1

f(p_{k-1})=kp_k+(n+1)(n+k-2)p_{k-1}

Problem Statement