Sure, let's analyze the expression step-by-step to determine which statements are true:
Given expression:
[tex]\[
6x^3 - 8x^2 - \frac{40}{y} + 21
\][/tex]
First, let's break it down to identify its components:
1. Term identification:
- The expression consists of several parts separated by addition or subtraction signs.
- The parts are:
- [tex]\( 6x^3 \)[/tex]
- [tex]\( -8x^2 \)[/tex]
- [tex]\( -\frac{40}{y} \)[/tex]
- [tex]\( +21 \)[/tex]
Thus, we see that there are four distinct terms in the expression:
- [tex]\( 6x^3 \)[/tex]
- [tex]\( -8x^2 \)[/tex]
- [tex]\( -\frac{40}{y} \)[/tex]
- [tex]\( 21 \)[/tex]
2. Examine each statement:
- Statement A: There are three terms.
- This statement is false because, as we identified, the expression has four terms.
- Statement B: The entire expression is a difference.
- This statement is false. Although the expression includes subtractions, it also includes addition (specifically the constant term [tex]\( +21 \)[/tex]). Thus, it is not purely a difference.
- Statement C: The term [tex]\( -\frac{40}{y} \)[/tex] is a ratio.
- This statement is true. The term [tex]\( \frac{40}{y} \)[/tex] (with the negative sign) indeed represents a ratio or a fraction.
- Statement D: There are four terms.
- This statement is true because we have identified four distinct terms in the expression.
From this analysis, the two true statements are:
- Statement C: The term [tex]\( -\frac{40}{y} \)[/tex] is a ratio.
- Statement D: There are four terms.
Summarizing, the correct statements regarding the expression [tex]\( 6x^3 - 8x^2 - \frac{40}{y} + 21 \)[/tex] are:
- The term [tex]\( -\frac{40}{y} \)[/tex] is a ratio (Statement C).
- There are four terms (Statement D).
These analyses are consistent with the accurate understanding of the expression provided.
