Give examples of polynomials $p(x), g(x), q(x)$ and $r(x)$ which satisfy the division algorithm and $\operatorname{deg} r(x) = 0$.

(A) According to the division algorithm,if $p(x)$ and $g(x)$ are two polynomials with $g(x) \neq 0$,then we can find polynomials $q(x)$ and $r(x)$ such that $p(x) = g(x) \times q(x) + r(x)$,where $r(x) = 0$ or $\operatorname{deg} r(x) < \operatorname{deg} g(x)$.
The degree of a polynomial is the highest power of the variable in the polynomial.
For $\operatorname{deg} r(x) = 0$,the remainder must be a non-zero constant.
Let us consider the division of $x^3 + 1$ by $x^2$.
Here,$p(x) = x^3 + 1$,$g(x) = x^2$.
Performing the division: $(x^3 + 1) \div x^2$ gives quotient $q(x) = x$ and remainder $r(x) = 1$.
Clearly,the degree of $r(x) = 1$ is $0$.
Checking the division algorithm:
$p(x) = g(x) \times q(x) + r(x)$
$x^3 + 1 = (x^2) \times x + 1$
$x^3 + 1 = x^3 + 1$
Thus,the division algorithm is satisfied.