Question

Which one of the following can be an example of a cubic polynomial?
$\begin{align}
  &A.\;{x^2} - x \\
  &B.\;{x^4} + {x^3} \\
  &C.\;{x^3} = 0 \\
  &D.\;{x^5} + {x^4} + {x^3} + {x^2} \\
\end{align} $

Answer 1

Hint: This problem requires the concept of cubic equations and polynomials. A cubic equation is an equation with degree 3, that is, the highest power of x in the equation is 3. In the question, we will check each and every option which satisfies the condition that the highest power of x in the equation is 3.

Complete step-by-step answer:
We need to find the cubic polynomials among the four options above, that is, the equation which has a degree 3.
In option A, we can simply check the degree of the polynomial. It has two terms, one with a degree 2 and the other with degree 1, this means that the overall equation has a degree of 2, so it is a quadratic equation.
${x^2} - x$
Here the degree of the polynomial is 2, hence this is not a cubic polynomial. This option is incorrect.
In option B,again we can simply check the degrees of the individual terms, and decide the overall degree of the equation. It also has two terms, one with a degree 4 and the other with degree 3, this means that the overall equation has a degree of 4, so it is a biquadratic equation.
${x^4} + {x^3}$
Now, it is clearly visible that this polynomial is not cubic and has a degree of 4. Hence, this option is also incorrect.
In option C, the given polynomial is already simplified, that is, ${x^3} = 0$ and is a cubic polynomial. This option satisfies the condition for a cubic polynomial, hence it is a correct option.

In option D, the equation has 4 terms, which have a degree of 5, 4, 3 and 2 respectively. Hence the overall degree is 5 and this option is incorrect as well.
${x^5} + {x^4} + {x^3} + {x^2}$
Hence, only option C satisfies the conditions of a cubic polynomial, therefore it is the correct option.

Note: A common mistake in this problem is that the students may mark option B and D is correct. This happens quite often in a hurry. In these two options, the polynomial seems to be cubic, due to the presence of a term with degree of 3. But the overall degree of those expressions is different. So we should look for polynomials which satisfy the conditions of a cubic polynomial only.