Question

In the following cases, use the remainder theorem and find the remainder when $p\left(x\right)$ is divided by $g\left(x\right)$.
$p\left(x\right)=x^3+3x^2-5x+8g\left(x\right)=x-3$

Answer 1

The Remainder Theorem states that when we divide a polynomial $p\left(x\right)$ by any factor $(x-a)$; which is not necessarily a factor of the polynomial; we will obtain a new smaller polynomial and a remainder, and this remainder is the value of $p\left(x\right)$ at $x=a$, that is $p\left(a\right)$.

Here, it is given that the polynomial $p\left(x\right)=x^3+3x^2-5x+8$ and the factor is $g\left(x\right)=x-3$, therefore, by remainder theorem, the remainder is $p\left(3\right)$ that is:

$p\left(3\right)=3^3+(3\times 3^2)-(5\times 3)+8=27+(3\times 9)-15+8=27+27-15+8=62-15=47$

Hence, the remainder is $r\left(x\right)=p\left(3\right)=47$.