Question

In the following case, use the remainder theorem and find the remainder when $p\left(x\right)$ is divided by $g\left(x\right)$.
$p\left(x\right)=2x^4-5x^2+15x-6g\left(x\right)=x-2$

Answer 1

The Remainder Theorem states that when you divide a polynomial $p\left(x\right)$ by any factor $(x-a)$; which is not necessarily a factor of the polynomial; you 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 $p\left(x\right)=2x^4-5x^2+15x-6$ and the factor is $g\left(x\right)=x-2$, therefore, by remainder theorem, the remainder is $p\left(2\right)$ that is:

$p\left(2\right)=(2\times 2^4)-(5\times 2^2)+(15\times 2)-6=(2\times 16)-(5\times 4)+30-6=32-20+30-6=32+30-20-6$

$=62-26=36$

Hence, the remainder is $r\left(x\right)=p\left(2\right)=36$.