Let's analyze the given mathematical expression step-by-step to determine which statements are true:
The expression given is:
[tex]\[ 5x^3 - 6x^2 - \frac{25}{y} + 18 \][/tex]
Step 1: Understand the expression structure
The expression consists of the following parts:
1. \( 5x^3 \) — a cubic term in \( x \), with a coefficient of 5.
2. \( -6x^2 \) — a quadratic term in \( x \), with a coefficient of -6.
3. \( -\frac{25}{y} \) — a term involving \( y \), creating a ratio where -25 is divided by \( y \).
4. \( +18 \) — a constant term.
Step 2: Analyze each statement
Statement A: "The entire expression is a difference."
- To qualify as a difference, the expression should represent subtraction only. The expression includes both addition and subtraction (evident from the terms \( +18 \) and \( -\frac{25}{y} \)), so this expression is not purely a difference.
- Statement A is false.
Statement B: "The term \( -\frac{25}{y} \) is a ratio."
- A ratio involves division, and \( -\frac{25}{y} \) clearly indicates \(-25\) divided by \( y \).
- Statement B is true.
Statement C: "There are three terms."
- Let's count the terms in the expression. We have four distinct parts: \( 5x^3 \), \( -6x^2 \), \( -\frac{25}{y} \), and \( +18 \).
- Statement C is false.
Statement D: "There are four terms."
- As already mentioned, the expression includes four distinct parts: \( 5x^3 \), \( -6x^2 \), \( -\frac{25}{y} \), and \( +18 \).
- Statement D is true.
Conclusion
The true statements are:
- Statement B: The term \( -\frac{25}{y} \) is a ratio.
- Statement D: There are four terms.
Thus, the correct pair of true statements are:
[tex]\[ \boxed{\text{B and D}} \][/tex]
