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.

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Answer 2

Answer:-

Given:-

According to the division algorithm, dividend p(x) and divisor g(x) are two polynomials, where g(x)≠0. Then we can find the value of quotient q(x) and remainder r(x), with the help of below

given formula;

Dividend = Divisor × Quotient + Remainder

=> p(x) = g(x)×q(x)+r(x)

Where r(x) = 0 or degree of r(x)< degree of g(x).

Now let us proof the three given cases as per division algorithm by taking examples for each.

Solve:-

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

Degree of dividend is equal to degree of quotient, only when the divisor is a constant term.

Let us take an example, p(x) = 3x^2+3x+3 is a polynomial to

be divided by g(x) = 3.

So, (3x^2+3x+3)/3 = x^2 + x + 1 = q(x)

Thus, you can see, the degree of quotient q(x) = 2, which also equal to the degree of dividend p(x).

Hence, division algorithm is satisfied here.

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

Let us take an example, p(x) = x^2 + 3 is a polynomial to be

divided by g(x) = x – 1.

So, x^2 + 3 = (x – 1)×(x) + (x + 3(

Hence, quotient q(x) = x

Also, remainder r(x) = x + 3

Thus, you can see, the degree of quotient q(x) = 1, which is also equal to the degree of remainder r(x).

iii) deg r(x) = 0

The degree of remainder is 0 only when the remainder left after division algorithm is constant.

Let us take an example, p(x) = x^2 + 1 is a polynomial to be

divided by g(x) = x.

So, x^2 + 1 = (x) × (x) + 1

Hence, quotient q(x) = x

And, remainder r(x) = 1

Clearly, the degree of remainder here is 0.

Hence, division algorithm is satisfied here