Case-Based Question:Two Brothers, Anuj And Aman, Started A Business Together. They Decided To Share The Capital Depending Upon The Variable Expenditure. The Capital Of The Two Brothers Together Is Given By The Polynomial $p^2 + 10p + 21$. This

Introduction

In this case-based question, we are presented with a scenario where two brothers, Anuj and Aman, have started a business together. They have decided to share the capital depending upon the variable expenditure. The capital of the two brothers together is given by the polynomial $p^2 + 10p + 21$. This polynomial represents the total capital that the brothers have invested in their business. In this article, we will explore the concept of variable expenditure and how it affects the sharing of capital between the two brothers.

Understanding the Polynomial

The given polynomial $p^2 + 10p + 21$ represents the total capital that the brothers have invested in their business. To understand this polynomial, we need to factorize it. Factoring a polynomial involves expressing it as a product of simpler polynomials. In this case, we can factorize the polynomial as follows:

$$p^2 + 10p + 21 = (p + 7)(p + 3)$$

This factorization helps us understand the polynomial better. We can see that the polynomial has two roots, $p = -7$ and $p = -3$. These roots represent the values of $p$ for which the polynomial becomes zero.

Variable Expenditure and Capital Sharing

The variable expenditure in this scenario refers to the expenses that the brothers incur in their business. These expenses can vary depending on the circumstances of the business. The brothers have decided to share the capital depending upon the variable expenditure. This means that the capital will be divided in a way that takes into account the variable expenses.

To understand how the capital will be shared, we need to consider the factorization of the polynomial. The factorization $(p + 7)(p + 3)$ represents the total capital that the brothers have invested in their business. The roots of the polynomial, $p = -7$ and $p = -3$, represent the values of $p$ for which the polynomial becomes zero.

Calculating the Capital Sharing

To calculate the capital sharing, we need to consider the factorization of the polynomial. The factorization $(p + 7)(p + 3)$ represents the total capital that the brothers have invested in their business. The roots of the polynomial, $p = -7$ and $p = -3$, represent the values of $p$ for which the polynomial becomes zero.

Let's assume that the variable expenditure is represented by the value of $p$. We can use the factorization to calculate the capital sharing as follows:

• If $p = -7$, the capital sharing will be $(p + 7)(p + 3) = (-7 + 7)(-7 + 3) = 0 \times (-4) = 0$
• If $p = -3$, the capital sharing will be $(p + 7)(p + 3) = (-3 + 7)(-3 + 3) = 4 \times 0 = 0$
• If $p = 0$, the capital sharing will be $(p + 7)(p + 3) = (0 + 7)(0 + 3) = 7 \times 3 = 21$

From the above calculations, we can see that the capital sharing depends on the value of $p$. If $p = -7$ or $p = -3$, the capital sharing will be zero. However, if $p = 0$, the capital sharing will be 21.

Conclusion

In this case-based question, we were presented with a scenario where two brothers, Anuj and Aman, have started a business together. They have decided to share the capital depending upon the variable expenditure. The capital of the two brothers together is given by the polynomial $p^2 + 10p + 21$. We factorized the polynomial to understand it better and calculated the capital sharing based on the factorization.

The capital sharing depends on the value of $p$, which represents the variable expenditure. If $p = -7$ or $p = -3$, the capital sharing will be zero. However, if $p = 0$, the capital sharing will be 21. This shows that the capital sharing is directly related to the variable expenditure and depends on the value of $p$.

Final Thoughts

In conclusion, this case-based question has helped us understand the concept of variable expenditure and how it affects the sharing of capital between two brothers. The polynomial $p^2 + 10p + 21$ represents the total capital that the brothers have invested in their business. We factorized the polynomial to understand it better and calculated the capital sharing based on the factorization.

The capital sharing depends on the value of $p$, which represents the variable expenditure. If $p = -7$ or $p = -3$, the capital sharing will be zero. However, if $p = 0$, the capital sharing will be 21. This shows that the capital sharing is directly related to the variable expenditure and depends on the value of $p$.

References

• [1] Algebraic Expressions and Equations. (n.d.). Retrieved from https://www.khanacademy.org/math/algebra
• [2] Polynomials. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/polynomials.html

Related Topics

• [1] Algebraic Expressions and Equations
• [2] Polynomials
• [3] Variable Expenditure and Capital Sharing

Keywords

• Case-based question
• Variable expenditure
• Capital sharing
• Polynomial
• Algebraic expressions and equations
• Polynomials
• Algebra
• Mathematics
  

Introduction

In our previous article, we explored the concept of variable expenditure and how it affects the sharing of capital between two brothers, Anuj and Aman. They have started a business together and decided to share the capital depending upon the variable expenditure. The capital of the two brothers together is given by the polynomial $p^2 + 10p + 21$. In this Q&A article, we will answer some of the most frequently asked questions related to this case-based question.

Q1: What is the significance of the polynomial $p^2 + 10p + 21$ in this scenario?

A1: The polynomial $p^2 + 10p + 21$ represents the total capital that the brothers have invested in their business. It is a quadratic equation that takes into account the variable expenditure.

Q2: How does the factorization of the polynomial help us understand the capital sharing?

A2: The factorization of the polynomial $(p + 7)(p + 3)$ helps us understand the capital sharing by showing that the roots of the polynomial, $p = -7$ and $p = -3$, represent the values of $p$ for which the polynomial becomes zero.

Q3: What happens to the capital sharing if $p = -7$ or $p = -3$?

A3: If $p = -7$ or $p = -3$, the capital sharing will be zero. This is because the polynomial becomes zero at these values of $p$.

Q4: What happens to the capital sharing if $p = 0$?

A4: If $p = 0$, the capital sharing will be 21. This is because the polynomial becomes 21 at this value of $p$.

Q5: How does the variable expenditure affect the capital sharing?

A5: The variable expenditure affects the capital sharing by changing the value of $p$. If the variable expenditure is high, the value of $p$ will be high, and the capital sharing will be affected accordingly.

Q6: Can you explain the concept of variable expenditure in more detail?

A6: The variable expenditure refers to the expenses that the brothers incur in their business. These expenses can vary depending on the circumstances of the business. The variable expenditure is represented by the value of $p$ in the polynomial.

Q7: How does the capital sharing depend on the value of $p$?

A7: The capital sharing depends on the value of $p$ by changing the value of the polynomial. If the value of $p$ is high, the capital sharing will be affected accordingly.

Q8: Can you provide an example of how the capital sharing would work in practice?

A8: Let's say the variable expenditure is $1000. The value of $p$ would be 1000. The capital sharing would be $(1000 + 7)(1000 + 3) = 1007 \times 1003 = 1013011$. This shows how the capital sharing would work in practice.

Q9: How does the capital sharing relate to the concept of algebraic expressions and equations?

A9: The capital sharing relates to the concept of algebraic expressions and equations by using the polynomial to represent the total capital that the brothers have invested in their business. The algebraic expressions and equations are used to solve for the value of $p$ and determine the capital sharing.

Q10: Can you provide a summary of the key points related to the capital sharing?

A10: The key points related to the capital sharing are:

• The capital sharing depends on the value of $p$.
• The value of $p$ represents the variable expenditure.
• The capital sharing is affected by the value of $p$.
• The polynomial $p^2 + 10p + 21$ represents the total capital that the brothers have invested in their business.
• The factorization of the polynomial helps us understand the capital sharing.

Conclusion

In this Q&A article, we have answered some of the most frequently asked questions related to the case-based question of two brothers and the variable expenditure business. We have explained the concept of variable expenditure and how it affects the capital sharing. We have also provided examples and summaries to help illustrate the key points related to the capital sharing.

Final Thoughts

In conclusion, the capital sharing is a complex concept that depends on the value of $p$, which represents the variable expenditure. The polynomial $p^2 + 10p + 21$ represents the total capital that the brothers have invested in their business. The factorization of the polynomial helps us understand the capital sharing by showing that the roots of the polynomial, $p = -7$ and $p = -3$, represent the values of $p$ for which the polynomial becomes zero.

References

• [1] Algebraic Expressions and Equations. (n.d.). Retrieved from https://www.khanacademy.org/math/algebra
• [2] Polynomials. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/polynomials.html

Related Topics

• [1] Algebraic Expressions and Equations
• [2] Polynomials
• [3] Variable Expenditure and Capital Sharing

Keywords

• Case-based question
• Variable expenditure
• Capital sharing
• Polynomial
• Algebraic expressions and equations
• Polynomials
• Algebra
• Mathematics