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

(ii)deq q(x)=deq r(x)

(iii)deg r(x)=0

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\[ \Rightarrow \] Dividend=Divisor × Quotient +Remainder

Here let us represent, p(x) =dividend=the number to be divided

g(x)=Divisor=the number by which dividend is divided

q(x)=quotient

And r(x)=remainder

Now we have to give examples of polynomials such that the division algorithm

\[ \Rightarrow \] Dividend=Divisor × Quotient +Remainder

Is satisfies and the given conditions are also satisfied

(i)Let us assume the division of $2x + 4$ by $2$

Then p(x) =$2x + 4$

g(x)= $2$

q(x)=$x + 2$

and r(x)=$0$

On using the division algorithm

\[ \Rightarrow \] Dividend=Divisor × Quotient +Remainder

$ \Rightarrow $ p(x)=g(x) × q(x) +r(x)

On putting the given values we get,

$ \Rightarrow 2x + 4 = 2 \times \left( {x + 2} \right) + 0$

On solving we get,

$ \Rightarrow 2x + 4 = 2x + 4$

Hence the division algorithm is satisfied.

And here the degree of p(x) =$1$ =degree of q(x)

Hence (i) condition is also satisfied.

(ii) Let us assume the division of ${x^3} + x$ by${x^2}$

Then here, p(x) =${x^3} + x$

g(x)= ${x^2}$

q(x)=$x$

r(x)=$x$

It is clear that the degree of q(x)=1 and

Degree of r(x)=1

On using the division algorithm

\[ \Rightarrow \] Dividend=Divisor × Quotient +Remainder

$ \Rightarrow $ p(x)=g(x) × q(x) +r(x)

On putting the given values we get,

$ \Rightarrow {x^3} + x = \left( {{x^2} \times x} \right) + x$

On solving we get,

$ \Rightarrow {x^3} + x = {x^3} + x$

Hence the division algorithm is satisfied.

And here the degree of r(x) =$1$ =degree of q(x)

Hence (ii) condition is also satisfied.

(iii)Let us assume the division of ${x^2} + 1$ by$x$

Then here, p(x) =${x^2} + 1$

g(x)= $x$

q(x)=$x$

r(x)=$1$

It is clear that the degree of r(x) =$0$ and

On using the division algorithm

\[ \Rightarrow \] Dividend=Divisor × Quotient +Remainder

$ \Rightarrow $ p(x)=g(x) × q(x) +r(x)

On putting the given values we get,

$ \Rightarrow {x^2} + 1 = \left( {x \times x} \right) + 1$

On solving we get,

$ \Rightarrow {x^2} + 1 = {x^2} + 1$

Hence the division algorithm is satisfied.

And here the degree of r(x) =$0$

Hence (iii) condition is also satisfied.