8. Find the remainders of dividing the polynomial

$$
f(x)=x^{243}+x^{81}+x^{27}+x^{9}+x^{3}+1
$$

by a) $x-1$; b) $x^{2}-1$.
$\triangle$ a) By the Remainder Theorem, the remainder from dividing the given polynomial by $x-1$ is $f(1)=6$.

b) It is possible to divide the given polynomial by $x^{2}-1$ using long division, but this is a lengthy method. It is simpler to isolate terms from the sum that are divisible by $x^{2}-1$:

$$
f(x)=\left(x^{243}-x\right)+\left(x^{81}-x\right)+\left(x^{27}-x\right)+\left(x^{9}-x\right)+\left(x^{3}-x\right)+(5 x+1)
$$

Each of the first five terms in the last sum is divisible by $x^{2}-1$ (think about why); therefore, the remainder from dividing $f(x)$ by $x^{2}-1$ is the sixth term $5 x+1$.

Answer: a) 6 ; b) $5 x+1$.