**Solution:**

According to the division algorithm, if p(x) and g(x) are two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x)

Degree of a polynomial is the highest power of the variable in the polynomial.

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

Degree of quotient will be equal to degree of dividend when divisor is constant ( i.e., when any polynomial is divided by a constant).

Let us assume the division of 6x^{2} + 2x + 2 by 2.

Here, p(x) = 6x^{2} + 2x + 2

g(x) = 2

q(x) = 3x^{2} + x + 1and 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)

6x^{2} + 2x + 2 = (2) (3x^{2} + x + 1) + 0

Thus, 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

Clearly, the 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

Thus, 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^{3} + 1 by x^{2}.

Here, p(x) = x^{3} + 1 g(x) = x^{2} q(x) = x and r(x) = 1

Clearly, the degree of r(x) is 0. Checking for division algorithm,

p(x) = g(x) × q(x) + r(x) x^{3} + 1 = (x^{2} ) × x + 1 x^{3} + 1 = x^{3} + 1

Thus, the division algorithm is satisfied.