Question

What is the remainder theorem?

Answer 1

Remainder theorem:

If $p\left(x\right)$ be any polynomial and $\text{'}a\text{'}$ be any real number. Then, when $p\left(x\right)$ is divided by the linear polynomial $x–a$, then the remainder will be $p\left(a\right)$.

Example:

Let polynomial is $p\left(x\right)=x^2+6x+4$ and it is divided by a linear polynomial $x-2$.

So, the remainder will be $p\left(2\right)$ which is computed as:

$$\begin{gathered} p\left(2\right)=2^2+(6\times 2)+4 \\ \Rightarrow p\left(2\right)=4+12+4 \\ \Rightarrow p\left(2\right)=20 \end{gathered}$$

Hence, If $p\left(x\right)$ is divided by the linear polynomial $(x-a)$, then the remainder is $p\left(a\right)$.