Sum of all coefficients of polynomial $P(x)$ is equal with $2$ . Also the sum of coefficients which are at odd exponential in $x^k$ are equal to sum of coefficients which are at even exponential in $x^k$ . Find the residue of polynomial $P(x)$ when it is divide by $x^2-1$ .
Problem
Source:
Tags: polynomial
claserken
09.01.2017 18:36
We have P(-1)=0 And P(1)=2
Not_a_Username
11.01.2017 02:12
^That is very helpful
AIME12345
11.01.2017 02:13
claserken wrote: We have $P(-1)=0$ And $P(1)=2$ FiXeD LaTeX. Very helpful, thank you for restating these two lines: dangerousliri wrote: Sum of all coefficients of polynomial $P(x)$ is equal with $2$ . dangerousliri wrote: Also the sum of coefficients which are at odd exponential in $x^k$ are equal to sum of coefficients which are at even exponential in $x^k$ . Now let's solve the problem.
rafayaashary1
11.01.2017 02:22
claserken wrote: We have P(-1)=0 And P(1)=2 Continuing, the remainder is $R(x)$ such that $P(x)=(x^2-1)Q(x)+R(x)$. Clearly $R(x)$ is linear, so by plugging in our two points we obtain $\boxed{x+1}$