Question

Which of the following is a constant polynomial?
$
  {\text{A}}{\text{. p}}\left( {\text{x}} \right) = {{\text{x}}^1} \\
  {\text{B}}{\text{. p}}\left( {\text{x}} \right) = {{\text{x}}^0} \\
  {\text{C}}{\text{. p}}\left( {\text{x}} \right) = {{\text{x}}^2} \\
  {\text{D}}{\text{. p}}\left( {\text{x}} \right) = {{\text{x}}^3} \\
$

Answer 1

Hint – A constant polynomial is a polynomial whose output value is independent of its input value. We check all the given options individually to find which option gives out a RHS value equal to a constant.

Complete step-by-step answer:

Given Data, Constant Polynomial
${\text{p}}\left( {\text{x}} \right) = {{\text{x}}^1}$
The RHS cannot be a constant as it is directly proportional to x. It is not a constant polynomial.

${\text{p}}\left( {\text{x}} \right) = {{\text{x}}^0}$
We know that any number or a variable to the power of zero is 1, i.e. ${{\text{a}}^0} = 1$, where a is any number.
Hence the equation becomes ${\text{p}}\left( {\text{x}} \right) = 1$
The RHS is a constant 1, hence it is in the form of a constant polynomial.

${\text{p}}\left( {\text{x}} \right) = {{\text{x}}^2}$
The RHS cannot be a constant as it is directly proportional to ${{\text{x}}^2}$. It is not a constant polynomial.

${\text{p}}\left( {\text{x}} \right) = {{\text{x}}^3}$
The RHS cannot be a constant as it is directly proportional to ${{\text{x}}^3}$. It is not a constant polynomial.

Hence Option B is the correct answer.

Note – In order to solve this type of problems the key is to know the definition of a constant polynomial. Constant polynomials are also called 0 degree polynomials. The output of them does not depend on the input variable. The graph of a constant polynomial is a horizontal line. It does not have any roots unless it is a polynomial of the form P(x) = 0.