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Give examples of polynomials and , which satisfy the division algorithm and
(i) deg deg  (ii) deg deg  (iii) deg

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Answer

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

We know the formula,

Dividend = Divisor x quotient + Remainder


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

Let us assume the division of by .
Here,

and

Degree of and is the same i.e., .

Checking for division algorithm,



Hence, the division algorithm is satisfied.

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

Let us assume the division of by ,
Here, p(x) = , g(x) = , q(x) = x and r(x) = x

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

Checking for division algorithm,





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 by
Here, p(x) =
g(x) =
and

Degree of is

Checking for division algorithm,



Hence, the division algorithm is satisfied.

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