Question

use the remainder therom to find the remainder when x^4-3x^2-9x+4 is divided by x-3

Answer 1

If we can write the given polynomial as

$\underbrace{p(x)}_{\text{given}}=q(x)(x-3)+r(x)$

that is, write $p(x)$ as the product of two factors plus some remainder term, then we can simply plug in $x=3$ to determine the remainder term because that would force the $q(x)$ term to vanish.

$p(x)=x^4-3x^2-9x+4\implies p(3)=31$

so the remainder upon dividing $p(x)$ by $x-3$ is 31.