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 $DegP\left(X\right)=DegQ\left(X\right)$.

Answer 1

Writing the required example:

We know the formula:

$$\begin{gathered} Dividend=Divisor\times quotient+Remainder \\ p\left(x\right)=g\left(x\right)\times q\left(x\right)+r\left(x\right) \end{gathered}$$

Let us take the division of $4x^2$ by $2$.

Here $p\left(x\right)=4x^2$

$$\begin{gathered} g\left(x\right)=2 \\ q\left(x\right)=2x^2 \\ r\left(x\right)=0 \end{gathered}$$

The degree of $p\left(x\right)$and $q\left(x\right)$ are the same which is $2$.

Checking for a division algorithm,

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

$4x^2=2\left(2x^2\right)$

LHS = RHS

Hence, the division algorithm is satisfied.