Question

Which of the following expressions are not polynomials?
(i) $3x^3-2x+1$
(ii) $x^2+x\sqrt{x}-2$
(iii) $x^4+2x$
(iv) $4x^2+3x-5$
(v) $\sqrt{2x}+\sqrt{3}$

Answer 1

We know that a polynomial is a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a non-negative integral power (whole number).

(i) $3x^3-2x+1$ can be rewritten as:

$3x^3-2x+1x^0$

The powers of the variable $x$ in the above equation are $3,1$ and $0$ which are whole numbers.

Therefore, $3x^3-2x+1$ is a polynomial.

(ii) $x^2+x\sqrt{x}-2$ can be rewritten as:

$x^2+x{\left(x\right)}^{\frac{1}{2}}-2x^0=x^2+{\left(x\right)}^{\frac{1}{2}+1}-2x^0=x^2+{\left(x\right)}^{\frac{3}{2}}-2x^0$

The powers of the variable $x$ in the above equation are $2,\frac{3}{2}$ and $0$, where$\frac{3}{2}$ is not a whole number.

Therefore, $x^2+x\sqrt{x}-2$ is not a polynomial.

(iii) $x^4+2x$ can be rewritten as:

$x^4+2x^1$

The powers of the variable $x$ in the above equation are $4$ and $1$ which are whole numbers.

Therefore, $x^4+2x$ is a polynomial.

(iv) $4x^2+3x-5$ can be rewritten as:

$4x^2+3x-5x^0$

The powers of the variable $x$ in the above equation are $2,1$ and $0$ which are whole numbers.

Therefore, $4x^2+3x-5$ is a polynomial.

(v) $\sqrt{2x}+\sqrt{3}$ can be rewritten as:

${\left(2x\right)}^{\frac{1}{2}}+\sqrt{3}{x}^0$

The powers of the variable $x$ in the above equation are $\frac{1}{2}$ and $0$, where$\frac{1}{2}$ is not a whole number.

Therefore, $\sqrt{2x}+\sqrt{3}$ is not a polynomial.