Choose Two Statements That Are True For This Expression: 6 X 3 − 8 X 2 − 40 Y + 21 6x^3 - 8x^2 - \frac{40}{y} + 21 6 X 3 − 8 X 2 − Y 40 ​ + 21 A. The Term − 40 Y -\frac{40}{y} − Y 40 ​ Is A Ratio. B. There Are Four Terms. C. There Are Three Terms. D. The Entire Expression Is A Difference.

The given expression is $6x^3 - 8x^2 - \frac{40}{y} + 21$. To determine which of the given statements are true, we need to analyze each statement individually.

Statement A: The term $-\frac{40}{y}$ is a ratio.

A ratio is a comparison of two numbers. In this case, the term $-\frac{40}{y}$ is a fraction, which represents a comparison between two numbers. The numerator is $-40$ and the denominator is $y$. Therefore, statement A is true.

Statement B: There are four terms.

To determine the number of terms in the expression, we need to count the individual parts. The expression is $6x^3 - 8x^2 - \frac{40}{y} + 21$. There are four individual parts: $6x^3$, $-8x^2$, $-\frac{40}{y}$, and $21$. Therefore, statement B is true.

Statement C: There are three terms.

This statement is contradictory to statement B. If there are four terms, then there cannot be three terms. Therefore, statement C is false.

Statement D: The entire expression is a difference.

A difference is the result of subtracting one number from another. In this case, the expression is $6x^3 - 8x^2 - \frac{40}{y} + 21$. The presence of the $+$ sign before the last term indicates that the expression is not a difference. Therefore, statement D is false.

Conclusion

In conclusion, the two true statements are:

• A. The term $-\frac{40}{y}$ is a ratio.
• B. There are four terms.

The other two statements are false. Statement C is false because there are four terms, not three. Statement D is false because the expression is not a difference.

Understanding the Components of the Expression

The given expression is $6x^3 - 8x^2 - \frac{40}{y} + 21$. Let's break down the components of the expression:

• Term 1: $6x^3$
• This term is a polynomial expression with a degree of 3.
• The coefficient of the term is 6.
• The variable is $x$.
• Term 2: $-8x^2$
• This term is a polynomial expression with a degree of 2.
• The coefficient of the term is $-8$.
• The variable is $x$.
• Term 3: $-\frac{40}{y}$
• This term is a fraction with a numerator of $-40$ and a denominator of $y$.
• The variable is $y$.
• Term 4: $21$
• This term is a constant.

Analyzing the Expression

The given expression is $6x^3 - 8x^2 - \frac{40}{y} + 21$. Let's analyze the expression:

• Degree of the expression: The highest degree of the polynomial terms is 3, which is the degree of the term $6x^3$. Therefore, the degree of the expression is 3.
• Coefficients: The coefficients of the polynomial terms are 6 and $-8$. The coefficient of the fraction term is $-40$ and the coefficient of the constant term is 21.
• Variables: The variables in the expression are $x$ and $y$.

Simplifying the Expression

The given expression is $6x^3 - 8x^2 - \frac{40}{y} + 21$. Let's simplify the expression:

• Combine like terms: There are no like terms in the expression, so no combination is possible.
• Simplify fractions: The fraction term is $-\frac{40}{y}$. This term cannot be simplified further.
• Simplify constants: The constant term is 21. This term cannot be simplified further.

Conclusion

In conclusion, the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$ cannot be simplified further. The expression is already in its simplest form.

Real-World Applications

The given expression is $6x^3 - 8x^2 - \frac{40}{y} + 21$. Let's consider some real-world applications of the expression:

• Physics: The expression can be used to model the motion of an object under the influence of gravity.
• Engineering: The expression can be used to design and optimize the performance of mechanical systems.
• Economics: The expression can be used to model the behavior of economic systems and make predictions about future trends.

Conclusion

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Q: What is the degree of the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$?

A: The degree of the expression is 3, which is the highest degree of the polynomial terms.

Q: What are the coefficients of the polynomial terms in the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$?

A: The coefficients of the polynomial terms are 6 and $-8$.

Q: What is the coefficient of the fraction term in the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$?

A: The coefficient of the fraction term is $-40$.

Q: What is the coefficient of the constant term in the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$?

A: The coefficient of the constant term is 21.

Q: What are the variables in the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$?

A: The variables in the expression are $x$ and $y$.

Q: Can the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$ be simplified further?

A: No, the expression cannot be simplified further. It is already in its simplest form.

Q: What are some real-world applications of the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$?

A: The expression can be used to model and analyze complex systems in various fields, including physics, engineering, and economics.

Q: Is the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$ a difference?

A: No, the expression is not a difference. The presence of the $+$ sign before the last term indicates that the expression is not a difference.

Q: How many terms are in the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$?

A: There are four terms in the expression.

Q: Is the term $-\frac{40}{y}$ a ratio?

A: Yes, the term $-\frac{40}{y}$ is a ratio.

Q: Is the term $-\frac{40}{y}$ a fraction?

A: Yes, the term $-\frac{40}{y}$ is a fraction.

Q: What is the numerator of the fraction term $-\frac{40}{y}$?

A: The numerator of the fraction term is $-40$.

Q: What is the denominator of the fraction term $-\frac{40}{y}$?

A: The denominator of the fraction term is $y$.

Q: What is the value of the constant term in the expression $6x^3 - 8x^2 - \frac{40}{y} + 21$?

A: The value of the constant term is 21.