#### Question

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

deg r(x) = 0

#### 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.

deg r(x) = 0

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

Let us assume the division of x^{3}+ 1by 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