Alright, let's carefully analyze the given expression step-by-step:
[tex]\[
5x^3 - 6x^2 - \frac{25}{y} + 18
\][/tex]
### Statement 1: The expression is a polynomial in [tex]\( x \)[/tex].
To qualify as a polynomial in [tex]\( x \)[/tex], the expression should solely involve non-negative integer powers of [tex]\( x \)[/tex] without any other variables in the denominators or exponents. Let's examine each term:
- [tex]\( 5x^3 \)[/tex]: A term with a power of 3, which is a non-negative integer.
- [tex]\( 6x^2 \)[/tex]: A term with a power of 2, which is a non-negative integer.
- [tex]\( 18 \)[/tex]: A constant term, which can be considered a term with [tex]\( x^0 \)[/tex].
- [tex]\( -\frac{25}{y} \)[/tex]: This term introduces a fraction involving [tex]\( y \)[/tex] in the denominator, meaning it isn't a simple polynomial term in [tex]\( x \)[/tex].
Due to the presence of [tex]\(-\frac{25}{y}\)[/tex], the expression as a whole is not a polynomial in [tex]\( x \)[/tex].
### Statement 2: The expression is a rational function because it involves a polynomial in the numerator and the variable [tex]\( y \)[/tex] in the denominator.
A rational function is defined as the ratio or combination of two polynomials. In this case, if we consider the numerator and the possible denominator:
- Numerator: All terms that are not involving [tex]\( y \)[/tex], our expression can be seen as a polynomial with terms [tex]\( 5x^3 \)[/tex], [tex]\(-6x^2\)[/tex], and [tex]\( 18 \)[/tex].
- Denominator: The presence of [tex]\( y \)[/tex] in [tex]\(-\frac{25}{y}\)[/tex] can be interpreted as a division by a function of [tex]\( y \)[/tex], though not directly representing the entire expression’s denominator, it is enough to indicate a form of rational component.
Since [tex]\(\frac{25}{y}\)[/tex] makes part of the expression rational by nature, this entire expression can indeed be interpreted as a rational function.
### Conclusions:
- The expression is not a polynomial in [tex]\( x \)[/tex] because of the [tex]\(-\frac{25}{y}\)[/tex] term.
- The expression is a rational function since it incorporates [tex]\( \frac{25}{y} \)[/tex] making it a ratio involving different variables.
Thus, the true statements are:
1. The expression is not a polynomial in [tex]\( x \)[/tex].
2. The expression is a rational function because it incorporates [tex]\(\frac{25}{y}\)[/tex].
