Let's analyze each function to determine whether it is a polynomial function.
A. [tex]\( F(x) = \frac{3}{5} x^4 - 18 x^3 + x^2 - 10 x + 3.6 \)[/tex]
A polynomial function is any function that can be written in the form:
[tex]\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \][/tex]
where all the coefficients [tex]\( a_i \)[/tex] are real numbers and [tex]\( n \)[/tex] is a non-negative integer.
In this case, [tex]\( F(x) \)[/tex] is expressed as a sum of terms with non-negative integer powers of [tex]\( x \)[/tex]. Therefore, [tex]\( F(x) = \frac{3}{5} x^4 - 18 x^3 + x^2 - 10 x + 3.6 \)[/tex] is indeed a polynomial function.
B. [tex]\( F(x) = 3 x^3 - 9 \)[/tex]
Again, this function is a sum of terms with non-negative integer powers of [tex]\( x \)[/tex]. Specifically, it is written as:
[tex]\[ 3 x^3 + 0 x^2 + 0 x - 9 \][/tex]
Since all the terms conform to the polynomial form with real coefficients, [tex]\( F(x) = 3 x^3 - 9 \)[/tex] is a polynomial function.
C. [tex]\( F(x) = -x^3 + 5 x^2 + 7 \sqrt{x} - 1 \)[/tex]
For a function to be a polynomial, all the exponents of [tex]\( x \)[/tex] must be non-negative integers. In [tex]\( F(x) \)[/tex], the term [tex]\( 7 \sqrt{x} \)[/tex] involves a fractional power of [tex]\( x \)[/tex] (specifically [tex]\( x^{1/2} \)[/tex]). This disqualifies it from being a polynomial function.
D. [tex]\( F(x) = 5.3 x^2 + 3 x - \frac{2}{x} + 6 \)[/tex]
Here, the term [tex]\( -\frac{2}{x} \)[/tex] involves a negative power of [tex]\( x \)[/tex] (specifically [tex]\( x^{-1} \)[/tex]). Because all exponents in a polynomial must be non-negative integers, this function is not a polynomial.
E. [tex]\( F(x) = 2 x^2 + 5 x - 3 \)[/tex]
This function is a sum of terms with non-negative integer powers of [tex]\( x \)[/tex]. It is written as:
[tex]\[ 2 x^2 + 5 x + (-3) \][/tex]
Since all the terms meet the criteria for a polynomial, [tex]\( F(x) = 2 x^2 + 5 x - 3 \)[/tex] is indeed a polynomial function.
Based on this analysis, the polynomial functions among the given options are:
[tex]\[ A, B, \text{ and } E \][/tex]
Therefore, the results are:
- [tex]\( A \)[/tex] is a polynomial function.
- [tex]\( B \)[/tex] is a polynomial function.
- [tex]\( C \)[/tex] is not a polynomial function.
- [tex]\( D \)[/tex] is not a polynomial function.
- [tex]\( E \)[/tex] is a polynomial function.
The final results in a binary form (where 1 represents polynomial and 0 represents non-polynomial) are:
[tex]\[ (1, 1, 0, 0, 1) \][/tex]
