Question

Find the highest power of $p$ when $81p^4-1$ is divided by $3p-1$.

Answer 1

We have

$81p^4-1=(3p{)}^4-1^4=(3p-1\left)\right(3p+1\left)\right\{(3p{)}^2+1\}$.

Now, $\frac{\left(81p^4\right)-1}{3p-1}=(3p+1)\left\{\right(3p{)}^2+1\}=(3p{)}^3+(3p{)}^2+(3p)+1$.

So the highest power of $p$ in the quotient is $3$.