Let's analyze the given mathematical expression step-by-step to determine the true statements about it.
The expression given is:
[tex]\[
5x^3 - 6x^2 - \frac{25}{y} + 18
\][/tex]
Step 1: Understanding the Terms
The expression can be broken down into four distinct parts:
1. [tex]\(5x^3\)[/tex]
2. [tex]\(-6x^2\)[/tex]
3. [tex]\(-\frac{25}{y}\)[/tex]
4. [tex]\(18\)[/tex]
These parts are separated by addition or subtraction signs, so each part is considered a separate term in the expression.
Statement A: The term [tex]\(-\frac{25}{y}\)[/tex] is a ratio.
- A ratio is defined as a relationship between two quantities, often expressed as a fraction.
- In this case, [tex]\(-\frac{25}{y}\)[/tex] is a fraction where [tex]\(-25\)[/tex] is divided by [tex]\(y\)[/tex], hence it can be regarded as a ratio.
Therefore, Statement A is true.
Statement B: There are four terms.
- As we identified earlier, the expression consists of the terms [tex]\(5x^3\)[/tex], [tex]\(-6x^2\)[/tex], [tex]\(-\frac{25}{y}\)[/tex], and [tex]\(18\)[/tex].
- This counts up to four distinct terms.
Therefore, Statement B is true.
Statement C: The entire expression is a difference.
- A difference is specifically defined as a subtraction between two quantities.
- The given expression involves both additions and subtractions:
- Addition of [tex]\(5x^3\)[/tex] and [tex]\(-6x^2\)[/tex], [tex]\(-\frac{25}{y}\)[/tex], and [tex]\(18\)[/tex].
Therefore, Statement C is false.
Statement D: There are three terms.
- As discussed, there are four terms in the expression.
Therefore, Statement D is false.
Consequently, after analyzing the given expression, the two statements that are true are:
A and B.
