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

(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 the degree of dividend when the divisor is constant.
Let us assume the division of 4x² by 2.
Here, p(x) = 4x², g(x) = 2, q(x) = 2x², 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)
4x² = 2(2x²)
Hence, the division algorithm is satisfied
(ii) 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
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 = x³ + x
Hence, the division algorithm is satisfied
(iii) deg r(x) = 0
Degree of remainder will be 0 when the remainder comes to a constant.
Let us assume the division of x⁴ + 1 by x³
Here, p(x) = x⁴ + 1, g(x) = x³, 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)
x⁴ + 1 = x³×x + 1
x⁴ + 1 = x⁴ + 1
Hence, the division algorithm is satisfied.