Question

1. kevin is given the function $f(x)=x^{2}-sqrt{3}x^{3}+x^{-4}+5x^{\frac{5}{4}}$. he believes that $f(x)$ is not a polynomial function for the following reasons:

i. the function contains a non - integral coefficient, $sqrt{3}$
ii. the function contains a negative power of $x$, - 4
iii. the function contains a rational power of $x$, $\frac{5}{4}$
iv. the functions does not contain a constant term.
which of kevins reasons about why $f(x)$ is not a polynomial functions are true?
a. i and ii
b. ii and iii
c. i, ii, and iii
d. i, iii, and iv

Explanation:

Step1: Recall polynomial function definition

A polynomial function in one - variable $x$ is of the form $f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0$, where $n$ is a non - negative integer and $a_i$ are real coefficients.

Step2: Analyze reason I

The coefficient of $x^3$ is $-\sqrt{3}$, which is a real number. A polynomial can have non - integral real coefficients. So, reason I is false.

Step3: Analyze reason II

The term $x^{-4}$ has a negative exponent. In a polynomial, the exponents of the variable must be non - negative integers. So, reason II is true.

Step4: Analyze reason III

The term $5x^{\frac{5}{4}}$ has a rational exponent. In a polynomial, the exponents of the variable must be non - negative integers. So, reason III is true.

Step5: Analyze reason IV

The absence of a constant term ($a_0$ when $x^0$) does not prevent a function from being a polynomial. For example, $f(x)=x^2+x$ is a polynomial without a non - zero constant term. So, reason IV is false.

Answer:

B. II and III