Question

Which of the following expressions are polynomials in one variable and which are not? State your reason.
A) \[3{{x}^{2}}-4x+15\]
B) \[{{y}^{2}}+2\sqrt{3}\]
C) \[3\sqrt{x}+\sqrt{2x}\]
D) \[x-\dfrac{4}{x}\]
E) \[{{x}^{12}}+{{y}^{3}}+{{t}^{50}}\]

Answer 1

Hint: To find the one variable polynomial, the equation must contain only one variable in terms of \[a{{x}^{n}}\] where \[n\] is the power as \[n\] is real, positive and non-fractional.

Complete step-by-step answer:
Let find one variable polynomial using the above criteria:
1. \[3{{x}^{2}}-4x+15\]
It’s a one variable polynomial as it contains a single \[a{{x}^{n}}\] term as \[3{{x}^{2}}\] in the equation.
2. \[{{y}^{2}}+2\sqrt{3}\]
It’s a one variable polynomial as it contains a single \[a{{x}^{n}}\] term as \[{{y}^{2}}\] in the equation.
3. \[3\sqrt{x}+\sqrt{2x}\]
It is not a one variable polynomial as the power of \[n\] in \[a{{x}^{n}}\] is \[\dfrac{1}{2},\left( 3{{x}^{\dfrac{1}{2}}} \right)\] which is fractional hence again not a one variable polynomial.
4. \[x-\dfrac{4}{x}\]
It is not a one variable polynomial as the variable \[\dfrac{4}{x}\] is negative in nature hence, contradicting the conditions.
5. \[{{x}^{12}}+{{y}^{3}}+{{t}^{50}}\]
It is not a one variable polynomial; it’s a three variable polynomial as there are three variables in form of \[a{{x}^{n}}\] as \[x,y,t\] and the value \[n\] follows all the conditions.

Note: Students may go wrong while solving the part number (4) as the first variable \[x\] can be placed as \[1.{{x}^{1}}\] but when finding one variable polynomial one must ensure that all the variables in the polynomial follow the condition for the power as \[n\] is real, positive and non-fractional.