Question

Give examples of polynomials $p\left(x\right),g\left(x\right),q\left(x\right)$ and $r\left(x\right)$, which satisfy the division algorithm and
(i) deg $p\left(x\right)=$deg $q\left(x\right)$ (ii) deg $q\left(x\right)=$deg $r\left(x\right)$ (iii) deg $r\left(x\right)=0$

Answer 1

(i) deg p(x) = deg q(x)

We know the formula,

Dividend = Divisor x quotient + Remainder

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

So here the degree of quotient will be equal to degree of dividend when the divisor is constant.

Let us assume the division of $4x^2$ by $2$.
Here, $p\left(x\right)=$$4x^2$
$g\left(x\right)=2$
$q\left(x\right)=$ $2x^2$ and $r\left(x\right)=0$

Degree of $p\left(x\right)$ and $q\left(x\right)$ is the same i.e., $2$.

Checking for division algorithm,
$p\left(x\right)=g\left(x\right)\times q\left(x\right)+r\left(x\right)$

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

Hence, the division algorithm is satisfied.

(ii) deg q(x) = deg r(x)

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

Here, p(x) = $x^3+x$, g(x) = $x^2$, q(x) = x and r(x) = x

Degree of q(x) and r(x) is the same i.e., 1.

Checking for division algorithm,

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

$x^3+x=x^2\times x+x$
$x^3+x=x^3+x$
Hence, the division algorithm is satisfied.

(iii) deg r(x) = 0

Degree of remainder will be 0 when remainder comes to a constant.
Let us assume the division of $x^4+1$ by $x^3$
Here, p(x) = $x^4+1$
g(x) = $x^3$
$q\left(x\right)=x$ and $r\left(x\right)=1$

Degree of $r\left(x\right)$ is $0.$

Checking for division algorithm,
$p\left(x\right)=g\left(x\right)\times q\left(x\right)+r\left(x\right)$
$x^4+1=x^3\times x+1$
$x^4+1=x^4+1$
Hence, the division algorithm is satisfied.