Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

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

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

Dividend=Divisor × Quotient +Remainder

 p(x)=g(x) × q(x) +r(x)

On putting the given values we get,

=> 2x + 4 = 2 x (x +2) + 0

On solving we get,

=> 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 x³ + x byx²

Then here,p(x) = x³+ x

g(x) = x²

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 devision algorithm

=> Dividend=Divisor × Quotient +Remainder

=> p(x)=g(x) × q(x) +r(x)

On putting the given values we get,

=> x³ + x = (x² * x) + x

on solving we get,

x³ + 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 x² + 1 byx

Then here,p(x) = x² + 1

g(x) = x

r(x) = 1

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

on using the devision algorithm

=> Dividend=Divisor × Quotient +Remainder

=> p(x)=g(x) × q(x) +r(x)

On putting the given values we get,

=> x² + 1 = (x * x)+ 1

x² + 1 = x² + 1

Hence,the division algorithm is satisfied.

And here the degree of r(x) = 0

Hence (iii) condition is also satisfied.

 Here you can also assume any other polynomial for division but it is necessary that the chosen dividend and divisor be such that the conditions of the questions are satisfied.