Question

Question 1
Which of the following expressions are polynomials? Justify your answer.
I. $8$
II. $\sqrt{3}{x}^2-2x$
III. $1-\sqrt{5x}$
IV. $\frac{1}{5x^{-2}}+5x+7$
V. $\frac{(x-2)(x-4)}{x}$
VI. $\frac{1}{x+1}$
VII. $\frac{1}{7}{a}^3-\frac{2}{\sqrt{3}}{a}^2+4a-7$
VIII. $\frac{1}{2x}$

Answer 1

i. Polynomial. Because, the exponent of the variable of 8 or $8x^0$ is 0 which is a whole number.

ii. Polynomial. Because, the exponent of the variable of $\sqrt{3}{x}^2-2$ is a whole number.

iii. Not a polynomial. Because, the exponent of the variable of $1-\sqrt{5x}or1-\sqrt{5}{x}^{\frac{1}{2}}$ is $\frac{1}{2}$ which is not a whole number.

iv. Polynomial. Because, the exponents of the variable of $\frac{1}{5x^{-2}}+5x+7=\frac{1}{5}{x}^2+5x+7$ are whole numbers.

v. Not a polynomial. Because, the exponent of the variable of $\frac{(x-2)(x-4)}{x}=\frac{x^2-6x+8}{x}=x-6+8x^{-1}$ is -1, which is not a whole number.

vi. Not a polynomial as the polynomial is expressed as $a_0+a_{1}x+a_2{x}^n$, where $a_0,a_1,a_2\cdots ,a_n$ are constants. Now, the given function is in the form $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ is a rational expression but not a polynomial.

vii. Polynomial. Because, the exponents of the variable of $\frac{1}{7}{a}^3-\frac{2}{\sqrt{3}}{a}^2+4a-7$ are whole numbers.

viii. Not a polynomial. Because, the exponent of the variable of $\frac{1}{2x}or\frac{1}{2}{x}^{-1}$ is -1, which is not a whole number.