What Is The Result Of The Following Operations?1. Dividing $x^2 + 3x + 1$ By $3x^2$ And Bringing Down $13x$.2. Subtracting $3x^4 + 9x^3 + 3x^2$ From The Dividend And Bringing Down $13x$.3. Multiplying

Introduction

In this article, we will explore the result of a series of mathematical operations involving polynomial division and subtraction. We will examine the steps involved in each operation and determine the final result.

Step 1: Dividing $x^2 + 3x + 1$ by $3x^2$ and Bringing Down $13x$

To begin, we need to divide the polynomial $x^2 + 3x + 1$ by $3x^2$. This can be done using long division or synthetic division. However, since we are bringing down $13x$, we can assume that the result of the division is a polynomial of the form $ax^2 + bx + c$.

Let's assume that the result of the division is $ax^2 + bx + c$. We can then multiply the divisor $3x^2$ by $ax^2 + bx + c$ and set it equal to the dividend $x^2 + 3x + 1$ plus the term $13x$ that we are bringing down.

(3x^2)(ax^2 + bx + c) = x^2 + 3x + 1 + 13x

Expanding the left-hand side of the equation, we get:

3ax^4 + 3bx^3 + 3cx^2 = x^2 + 3x + 1 + 13x

Simplifying the equation, we get:

3ax^4 + 3bx^3 + 3cx^2 - x^2 - 16x - 1 = 0

Now, we need to find the values of $a$, $b$, and $c$ that satisfy this equation.

Step 2: Subtracting $3x^4 + 9x^3 + 3x^2$ from the Dividend and Bringing Down $13x$

In this step, we need to subtract the polynomial $3x^4 + 9x^3 + 3x^2$ from the dividend $x^2 + 3x + 1 + 13x$. This can be done by combining like terms.

(x^2 + 3x + 1 + 13x) - (3x^4 + 9x^3 + 3x^2) = -3x^4 - 9x^3 - 2x^2 + 16x - 1

Now, we need to find the result of this subtraction.

Step 3: Multiplying the Result by $3x^2$

In this step, we need to multiply the result of the subtraction by $3x^2$. This can be done by multiplying each term in the result by $3x^2$.

(-3x^4 - 9x^3 - 2x^2 + 16x - 1)(3x^2) = -9x^6 - 27x^5 - 6x^4 + 48x^3 - 3x^2

Now, we need to find the final result.

Conclusion

In this article, we have explored the result of a series of mathematical operations involving polynomial division and subtraction. We have examined the steps involved in each operation and determined the final result.

The final result is:

-9x^6 - 27x^5 - 6x^4 + 48x^3 - 3x^2

This result is obtained by multiplying the result of the subtraction by $3x^2$.

References

• [1] "Polynomial Division" by Math Open Reference
• [2] "Subtraction of Polynomials" by Math Is Fun
• [3] "Multiplication of Polynomials" by Mathway

Note

The result of the operations is a polynomial of degree 6. The coefficients of the polynomial are integers. The polynomial can be factored as:

-9x^6 - 27x^5 - 6x^4 + 48x^3 - 3x^2 = -9x^2(x^4 + 3x^3 + 2x^2 - 16x + 1/3)

Introduction

In our previous article, we explored the result of a series of mathematical operations involving polynomial division and subtraction. We determined the final result and provided a detailed explanation of the steps involved. In this article, we will answer some frequently asked questions about the result of the operations.

Q: What is the degree of the final polynomial?

A: The degree of the final polynomial is 6.

Q: What are the coefficients of the final polynomial?

A: The coefficients of the final polynomial are integers.

Q: Can the final polynomial be factored?

A: Yes, the final polynomial can be factored as:

-9x^6 - 27x^5 - 6x^4 + 48x^3 - 3x^2 = -9x^2(x^4 + 3x^3 + 2x^2 - 16x + 1/3)

Q: What is the significance of the factorization?

A: The factorization of the final polynomial makes it easier to work with and can be used to simplify the polynomial.

Q: Can the final polynomial be used in real-world applications?

A: Yes, the final polynomial can be used in real-world applications such as engineering, physics, and computer science.

Q: How can the final polynomial be used in engineering?

A: The final polynomial can be used in engineering to model and analyze complex systems. For example, it can be used to model the behavior of a mechanical system or to analyze the stability of a control system.

Q: How can the final polynomial be used in physics?

A: The final polynomial can be used in physics to model and analyze complex phenomena. For example, it can be used to model the behavior of a particle in a potential field or to analyze the stability of a system.

Q: How can the final polynomial be used in computer science?

A: The final polynomial can be used in computer science to model and analyze complex algorithms. For example, it can be used to model the behavior of a recursive algorithm or to analyze the complexity of a computational problem.

Q: What are some common applications of polynomial division and subtraction?

A: Polynomial division and subtraction are used in a variety of applications, including:

• Engineering: to model and analyze complex systems
• Physics: to model and analyze complex phenomena
• Computer Science: to model and analyze complex algorithms
• Mathematics: to solve equations and inequalities
• Data Analysis: to analyze and visualize data

Conclusion

In this article, we have answered some frequently asked questions about the result of the operations involving polynomial division and subtraction. We have provided a detailed explanation of the steps involved and discussed the significance of the factorization of the final polynomial. We have also discussed some common applications of polynomial division and subtraction.

References

• [1] "Polynomial Division" by Math Open Reference
• [2] "Subtraction of Polynomials" by Math Is Fun
• [3] "Multiplication of Polynomials" by Mathway

Note

The result of the operations is a polynomial of degree 6. The coefficients of the polynomial are integers. The polynomial can be factored as:

-9x^6 - 27x^5 - 6x^4 + 48x^3 - 3x^2 = -9x^2(x^4 + 3x^3 + 2x^2 - 16x + 1/3)

This factorization can be used to simplify the polynomial and make it easier to work with.