Question

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

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

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

(iii) deg r(x) = 0

Answer 1

Answer:

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

We know the formula,

Dividend = Divisor x quotient + Remainder

p(x)=g(x)×q(x)+r(x)

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(x)=4x

2

g(x)=2

q(x)= 2x

2

and r(x)=0

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

Checking for division algorithm,

p(x)=g(x)×q(x)+r(x)

4x

2

=2(2x

2

)

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(x)=g(x)×q(x)+r(x)

x

3

+x=x

2

×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(x)=x and r(x)=1

Degree of r(x) is 0.

Checking for division algorithm,

p(x)=g(x)×q(x)+r(x)

x

4

+1=x

3

×x+1

x

4

+1=x

4

+1

Hence, the division algorithm is satisfied.