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) $x^{1000}-1$

(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) $\sqrt[2]{22}{x}^2-8$

(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) $x^{1000}-1$ is an expression having an 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) $1$. Clearly, 1 is a constant polynomial of degree 0.

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

(ix) $\frac{x^2}{2}-\frac{2}{x^2}$ . This is an expression having a negative integral power of x i.e. −2. So, it is not a polynomial.

(x) $\sqrt[2]{22}{x}^2-8$ is an expression having an 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) $\frac{1}{2x^2}$ is an expression having a 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 12 which is a fraction. Since it is an expression having a 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 32 which is a fraction. Since it is an expression having a fractional power of x, so, it is not a polynomial.

(xv) $2x^3+3x^2+\sqrt{x}-1$ . In this expression, one of the powers of x is 12 which is a fraction. Since it is an expression having a fractional power of x, so, it is not a polynomial.