Which of the following expressions are polynomials? In case of a polynomial, write its degree.
Question:

Which of the following expressions are polynomials? In case of a polynomial, write its degree.

(i) $x^{5}-2 x^{3}+x+\sqrt{3}$

(ii) $y^{3}+\sqrt{3} y$

(iii) $t^{2}-\frac{2}{5} t+\sqrt{5}$

(iv) $x^{100}-1$

(v) $\frac{1}{\sqrt{2}} x^{2}-\sqrt{2} x+2$

(vi) $x^{-2}+2 x^{-1}+3$

(vii) 1

(viii) $\frac{-3}{5}$

(ix) $\frac{x^{2}}{2}-\frac{2}{x^{2}}$

(x) $\sqrt[3]{2} x^{2}-8$

(xi) $\frac{1}{2 x^{2}}$

(xii) $\frac{1}{\sqrt{5}} x^{\frac{1}{2}}+1$

(xiii) $\frac{3}{5} x^{2}-\frac{7}{3} x+9$

(xiv) $x^{4}-x^{\frac{3}{2}}+x-3$

(xv) $2 x^{3}+3 x^{2}+\sqrt{x}-1$

 

Solution:

(i) $x^{5}-2 x^{3}+x+\sqrt{3}$ is an expression having only non-negative integral powers of $x .$ So, it is a polynomial. Also, the highest power of $x$ is 5, so, it is a polynomial of degree 5 .

(ii) $y^{3}+\sqrt{3} y$ is an expression having only non-negative integral powers of $y$. So, it is a polynomial. Also, the highest power of $y$ is 3 , so, it is a polynomial of degree $3 .$

(iii) $t^{2}-\frac{2}{5} t+\sqrt{5}$ is an expression having only non-negative integral powers of $t$. So, it is a polynomial. Also, the highest power of $t$ is 2 , so, it is a polynomial of degree $2 .$

(iv) $x^{100}-1$ is an expression having only non-negative integral power of $x$. So, it is a polynomial. Also, the highest power of $x$ is 100 , so, it is a polynomial of degree 100 .

(v) $\frac{1}{\sqrt{2}} x^{2}-\sqrt{2} x+2$ is an expression having only non-negative integral powers of $x$. So, it is a polynomial. Also, the highest power of $x$ is 2, so, it is a polynomial of degree 2 .

(vi) $x^{-2}+2 x^{-1}+3$ is an expression having negative integral powers of $x$. So, it is not a polynomial.

(vii) Clearly, 1 is a constant polynomial of degree $0 .$

(viii) Clearly, $-\frac{3}{5}$ is a constant polynomial of degree 0 .

(ix) $\frac{x^{2}}{2}-\frac{2}{x^{2}}=\frac{x^{2}}{2}-2 x^{-2}$

This is an expression having negative integral power of $x$ i.e. $-2$. So, it is not a polynomial.

(x) $\sqrt[3]{2} x^{2}-8$ is an expression having only non-negative integral power of $x$. So, it is a polynomial. Also, the highest power of $x$ is 2, so, it is a polynomial of degree 2 .

(xi) $\frac{1}{2 x^{2}}=\frac{1}{2} x^{-2}$ is an expression having negative integral power of $x .$ So, it is not a polynomial.

(xii) $\frac{1}{\sqrt{5}} x^{\frac{1}{2}}+1$

In this expression, the power of $x$ is $\frac{1}{2}$ which is a fraction. Since it is an expression having fractional power of $x$, so, it is not a polynomial.

(xiii) $\frac{3}{5} x^{2}-\frac{7}{3} x+9$ is an expression having only non-negative integral powers of $x$. So, it is a polynomial. Also, the highest power of $x$ is 2, so, it is a polynomial of degree $2 .$

(xiv) $x^{4}-x^{\frac{3}{2}}+x-3$

In this expression, one of the powers of $x$ is $\frac{3}{2}$ which is a fraction. Since it is an expression having fractional power of $x$, so, it is not a polynomial.

(xv) $2 x^{3}+3 x^{2}+\sqrt{x}-1=2 x^{3}+3 x^{2}+x^{\frac{1}{2}}-1$

In this expression, one of the powers of $x$ is $\frac{1}{2}$ which is a fraction. Since it is an expression having fractional power of $x$, so, it is not a polynomial.

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