Question

Give examples of polynomials $P\left(X\right),G\left(X\right),Q\left(X\right),andR\left(X\right)$, which satisfy the Division Algorithm and $DegQ\left(X\right)=DegR\left(X\right)$.

Answer 1

Writing the required example:

Let us assume the division of $x^3+x$ by $x^2$. Here

$$\begin{gathered} p\left(x\right)=x^3+x \\ g\left(x\right)=x^2 \\ q\left(x\right)=x \\ r\left(x\right)=x \end{gathered}$$

The degree of $q\left(x\right)$ and $r\left(x\right)$ is the same which is $1$.

Checking for a division algorithm,

$p\left(x\right)=g\left(x\right)\times q\left(x\right)+r\left(x\right)$

$$\begin{gathered} x^3+x=x^2\times x+x \\ {x}^3+x=x^3+x \end{gathered}$$

Since, LHS=RHS

Hence, an example satisfying the division algorithm and $DegQ\left(X\right)=DegR\left(X\right)$ is given.