Question

Which of the following expressions are polynomials? In case of a polynomial, write its degree.
(i) $x^5-2x^3+x+\sqrt{3}$
(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) $x^{-2}+2x^{-1}+3$
(vii) 1
(viii) $\frac{-3}{5}$
(ix) $\frac{x^2}{2}-\frac{2}{x^2}$
(x) $\sqrt[3]{2}{x}^2-8$
(xi) $\frac{1}{2x^2}$
(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) $2x^3+3x^2+\sqrt{x}-1$

Answer 1

(i) $x^5-2x^3+x+\sqrt{3}$ 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) $x^{-2}+2x^{-1}+3$ 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) $\frac{x^2}{2}-\frac{2}{x^2}=\frac{x^2}{2}-2x^{-2}$
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) $\frac{1}{2x^2}=\frac{1}{2}{x}^{-2}$ 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) $2x^3+3x^2+\sqrt{x}-1=2x^3+3x^2+x^{\frac{1}{2}}-1$
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.