## Related questions with answers

(a) Prove that there exist unique scalars $c_0,c_1,\dots,c_n$ such that

$f(-1)+f(-2)=c_0f(0)+c_1f(1)+\dots+c_nf(n)$

for every polynomial $f(x)$ in $\mathcal{P}_n$. (b) Find the scalars $c_0,c_1,c_2,c_3,$ and $c_4$ such that

$f(-1)+f(-2)=c_0f(0)+c_1f(1)+c_2f(2)+c_3f(3)+c_4f(4)$

for every polynomial $f(x)$ in $\mathcal{P}_4$.

Solution

Verified#### (a)

According to $\textit{Exercise 77}$, there exist unique scalars $c_0,c_1,\dots,c_n$ such that

$\int\limits_{-2}^{-1}f(x)dx=c_0f(0)+c_1f(1)+\dots+c_nf(n)$

for every polynomial $f(x)$ in $\mathcal{P}_n$. The expression on the left is a number determined uniquely by a polynomial $f(x)$, if we switch this expression with $f(-1)+f(-2)$, this is still determined uniquely by the polynomial $f(x)$ and it doesn't change the statement.

Therefore, there exist unique scalars $c_0,c_1,\dots,c_n$ such that

$f(-1)+f(-2)=c_0f(0)+c_1f(1)+\dots+c_nf(n)$

for every polynomial $f(x)$ in $\mathcal{P}_n$.

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