Which of the following is a quadratic polynomial in one variable?
$\begin{align}
  &A.\;\sqrt {2{x^3}} + 5 \\
  &B.\;2{x^2} + 2{x^{ - 2}} \\
  &C.\;{x^2} \\
  &D.\;2{x^2} + {y^2} \\
\end{align} $

Question

Which of the following is a quadratic polynomial in one variable?$\begin{align}  &A.\;\sqrt {2{x^3}}  + 5  \\  &B.\;2{x^2} + 2{x^{ - 2}}  \\  &C.\;{x^2}  \\  &D.\;2{x^2} + {y^2}  \\ \end{align} $

Vedantu Content Team · Accepted Answer

Hint: This problem requires the concept of quadratic equations and the equations in one variable. A quadratic equation is an equation with degree 2, that is, the highest power of x in the equation is 2. Also, an equation in one variable is the one which is dependent on only one variable. In the question, we will check each and every option which satisfies both the above equations.

Complete step-by-step answer:
We need to find the quadratic polynomials in one variable among the four options above, that is, the equation which has a degree 2 and has only a single variable.
In option A, we can simplify the square root to check the degree of the polynomial. Also, we can see that it is dependent on only a single variable, that is x.
$\begin{align}
   &= \sqrt {2{x^3}} + 5 \\
   &= \sqrt 2 {x^{\dfrac{3}{2}}} + 5 \\
\end{align} $
Here the degree of the polynomial is 1.5, hence this is not a quadratic polynomial. This option is incorrect.
In option B, it may seem that the polynomial is quadratic. But when we further simplify the polynomial, we can write that-
$\begin{align}
   &= 2{x^2} + 2{x^{ - 2}} \\
   &= 2{x^2} + \dfrac{2}{{{x^2}}} \\
   &= \dfrac{{2{x^4} + 2}}{{{x^2}}} \\
\end{align} $
Now, it is clearly visible that this polynomial is not quadratic. Hence, this option is also incorrect.
In option C, the given polynomial is already simplified, that is, ${x^2}$ and is a quadratic polynomial which is dependent on only a single variable. This option satisfies both the conditions, hence it is a correct option.
In option D, the equation is quadratic, but it is dependent on two variables x and y, hence this option is incorrect as well.
Hence, only option C satisfies both the conditions, 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 option B, the polynomial seems to be quadratic, but on simplification, it has a degree more than 2. Similarly in option D, the polynomial is quadratic, but has more than one variable. So we should look for polynomials which satisfy both the conditions.