Question

Which of the following is a cubic polynomial?
A. $p(x) = {x^3} - {3^3}$
B. $p(x) = x - 3$
C. $p(x) = 3$
D. $p(x) = {3^3}$

Answer 1

Hint:
We will check the highest power of the variable on which the polynomial is dependent, if the highest power of the variable is three, we would say the whole polynomial to be cubic. The highest power of the variable in a polynomial is also known as its degree. Hence, the polynomial with degree 3 would be our final answer.

Complete step by step solution:
Now, we will find out the highest power of the variable on which the polynomial is dependent(Degree) for each given choice.

For option1, the polynomial is given by
 $p(x) = {x^3} - {3^3}$
Here, the polynomial is dependent on x, Hence we will check for the highest power of x (Degree).

We can see that degree of option A is given by
 $ \Rightarrow {D_{{x^3} - {3^3}}} = 3$
Since the degree is 3, we can say that it is a cubic polynomial.

For option B, the polynomial is given by
 $p(x) = x - 3$
Here, the polynomial is dependent on x, Hence we will check for the highest power of x (Degree).

We can see that degree of option B is given by
 $ \Rightarrow {D_{x - 3}} = 1$
Since the degree is 1, we can say that it is NOT a cubic polynomial.

For option C, the polynomial is given by
 $p(x) = 3$
Here, the polynomial is dependent on x, Hence we will check for the highest power of x (Degree).

We can see that degree of option C is given b
 $ \Rightarrow {D_3} = 0$
Since the degree is 0, we can say that it is NOT a cubic polynomial.

For option D, the polynomial is given by
 $p(x) = {3^3}$
Here, the polynomial is dependent on x, Hence we will check for the highest power of x (Degree).

We can see that degree of option D is given by
 $ \Rightarrow {D_{{3^3}}} = 0$
Since the degree is 0, we can say that it is NOT a cubic polynomial.

Hence, the final answer is A.

Note:
The degree of a polynomial is the highest power of the variable on which the polynomial is dependent, If the question would have asked for a quadratic polynomial, we would have to select the polynomial whose degree is 2, And if the question would have asked for a biquadratic polynomial, we would have searched for a polynomial with degree 4.