Question:
By actual division, find the quotient and the remainder when $\left(x^{4}+1\right)$ is divided by $(x-1)$. Verify that remainder $=f(1)$.
Solution:
Let $f(x)=x^{4}+1$ and $g(x)=x-1$
Quotient $=x^{3}+x^{2}+x+1$
Remainder $=2$
Verification:
Putting $x=1$ in $f(x)$, we get
$f(1)=1^{4}+1=1+1=2=$ Remainder, when $f(x)=x^{4}+1$ is divided by $g(x)=x-1$