Question

Which of the following are not polynomials?
(i) $2-3x$
(ii) $\sqrt{2y^3}+\sqrt{3y}$
(iii) $x^2+x\sqrt{x}$
(iv) $x^2+\frac{7}{6}{x}^2-9$
(v) $u^{-1/2}+3u+2$

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) $2-3x$ can be rewritten as:

$2x^0-3x$

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

Therefore, $2-3x$ is a polynomial.

(ii) $\sqrt{2y^3}+\sqrt{3y}$ can be rewritten as:

${\left(2y\right)}^{\frac{3}{2}}+{\left(3y\right)}^{\frac{1}{2}}$

The powers of the variable $y$ in the above equation are $\frac{3}{2}$ and $\frac{1}{2}$ which are not whole numbers.

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

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

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

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

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

(iv) $x^2+\frac{7}{6}{x}^2-9$ can be rewritten as:

$x^2+\frac{7}{6}{x}^2-9x^0$

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

Therefore, $x^2+\frac{7}{6}{x}^2-9$ is a polynomial.

(v) $u^{-\frac{1}{2}}+3u+2$ can be rewritten as:

$u^{-\frac{1}{2}}+3u+2u^0$

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

Therefore, $u^{-\frac{1}{2}}+3u+2$ is not a polynomial.