Find the remainder from dividing the polynomial $P(x)=x^{81}+x^{27}+x^{9}+x^{3}+x$ by

a) $x-1$;

b) $x^{2}-1$.

#
a) $P(1)=5$.

b) First method. Let $P(x)=(x^2-1)Q(x) + ax + b$. Substituting $x=-1$, we get $b-a=-5$; substituting $x=1$, we get $a+b=5$.

From this, $b=0, a=5$.

Second method. $P(x)=x(x^{80}+x^{26}+x^{8}+x^{2}+1)$. By substituting $t=x^2$, the second factor transforms into the polynomial $Q(t)=t^{40}+t^{13}+t^{4}+t+1$, which, when divided by $t-1$, gives a remainder of $Q(1)=5$. Therefore, $P(x)$, when divided by $x^2-1$, gives a remainder of $x \cdot 5$.

## Answer

a) 5; b) $5x$.