Question

Which of the following is a cubic polynomial?
$\begin{align}
  & \left( a \right)p\left( x \right)={{y}^{3}}-27 \\
 & \left( b \right)p\left( y \right)={{x}^{3}}-27 \\
 & \left( c \right)p\left( x \right)={{x}^{3}}-27 \\
 & \left( d \right)p\left( y \right)=27 \\
\end{align}$

Answer 1

Hint: To solve the question given above, we will first find out what is a polynomial and what is the general form of writing a polynomial and what is a cubic polynomial. After finding these, we will check each option one by one the option which will satisfy the definition of cubic polynomial will be the answer for this question.

Complete step-by-step answer:
Before we solve this question, we must know what a polynomial is. A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication and non negative integer exponents of variables. The general representation of a polynomial in x is p(x) and its general form is:
$p\left( x \right)={{a}_{1}}{{x}^{n}}+{{a}_{2}}{{x}^{n-1}}+{{a}_{3}}{{x}^{n-2}}+..........{{a}_{n}}x+{{a}_{n+1}}$
When the value of n becomes 3, we get a cubic polynomial. Thus, the general form of cubic polynomial is: $p\left( x \right)=a{{x}^{3}}+b{{x}^{2}}+cx+d$
Now, we will check each option one by one:
Option (a): $p\left( x \right)={{y}^{3}}-27$. According to this option, we have a polynomial is x. For this polynomial to be cubic, the form $a{{x}^{3}}$ should be present and no term greater power of 3 should be present. But the power of x in the above polynomial is zero.
Option (b): $p\left( y \right)={{x}^{2}}-27$. According to this option, we have a polynomial in y. The term present in RHS should be of the form: $a{{y}^{3}}+b{{y}^{^{2}}}+cy+d$. But in RHS, it is not present.
Option (c): $p\left( x \right)={{x}^{3}}-27.$ According to this option, we have a polynomial in x. The term present in RHS should be of the form $a{{x}^{3}}+b{{x}^{2}}+cx+d$. In this case, this term matches with the option when we put $a=1,b=0,c=0\text{ and}\ \text{d=-27}$.
Option (d): $p\left( y \right)=27.$ In RHS, the maximum power of y is 0. But in cubic polynomials, the maximum power of y is 3. Hence it is not a cubic polynomial.
Thus, option (c) is correct.

Note: In power of x is any polynomial should be a positive integer and the polynomial should contain only the terms of power of x. Other functions cannot be present in this polynomial. For example:
$p\left( x \right)={{x}^{3}}+5x+{{\tan }^{-1}}\left( \dfrac{4x+2}{7} \right)+3$
The above given equation is not a polynomial because it contains a trigonometric factor.