Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

(i) $4 x^{2}-3 x+7$

(ii) $y^{2}+\sqrt{2}$

(iii) $3 \sqrt{t}+t \sqrt{2}$

(iv) $y+\frac{2}{y}$

(v) $x^{10}+y^{3}+t^{50}$
Solution:

(i) $4 x^{2}-3 x+7$

Yes, this expression is a polynomial in one variable $x$.

(ii) $y^{2}+\sqrt{2}$

Yes, this expression is a polynomial in one variable $y$.

(iii) $3 \sqrt{t}+t \sqrt{2}$

No. It can be observed that the exponent of variable $t$ in term $3 \sqrt{t}$ is $\frac{1}{2}$, which is not a whole number. Therefore, this expression is not a polynomial.

(iv) $y+\frac{2}{y}$

No. It can be observed that the exponent of variable $y$ in term $\frac{2}{y}$ is $-1$, which is not a whole number. Therefore, this expression is not a polynomial.

(v) $x^{10}+y^{3}+t^{50}$

No. It can be observed that this expression is a polynomial in 3 variables $x, y$, and $t .$ Therefore, it is not a polynomial in one variable.
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