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.

1. 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 6x² + 2x + 2 by 2

Here, p(x) = 6x² + 2x + 2

g(x) = 2

q(x) = 3x² + x + 1 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)

6x² + 2x + 2 = 2(3x² + x + 1) + 0

Thus, the division algorithm is satisfied.

2. deg q(x) = deg r(x)

Let us assume the division of x³ + x by x²

Here, p(x) = x³ + x,  g(x) = x² , 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)

x³ + x = x² * x + x = x³ + x

Thus, the division algorithm is satisfied.

3. deg r(x) = 0

Degree of remainder will be 0 when remainder comes to a constant.

Let us assume the division of x³ + 1 by x²

Here, p(x) = x³ + 1, g(x) = x² , 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)

x³ + 1 = x² * x + 1 = x³ + 1

Thus, the division algorithm is satisfied.