Rewrite The Following Division Problem So It Is Clear And Organized:Perform The Long Division:1. Divide The Polynomial X 3 + 3 X 2 + X X^3 + 3x^2 + X X 3 + 3 X 2 + X By X + 3 X + 3 X + 3 .2. Subtract X 3 + 2 X 2 X^3 + 2x^2 X 3 + 2 X 2 From The Result.3. Continue The Division Process To

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial skill to master, especially when working with complex equations and functions. In this article, we will rewrite the given division problem and provide a clear and organized solution.

Rewriting the Division Problem

The given problem involves performing long division on the polynomial $x^3 + 3x^2 + x$ by $x + 3$. To make the problem more manageable, we can break it down into smaller steps.

Step 1: Divide the Polynomial

To divide the polynomial $x^3 + 3x^2 + x$ by $x + 3$, we can use the long division method. We start by dividing the leading term of the dividend ($x^3$) by the leading term of the divisor ($x$), which gives us $x^2$.

x^3 + 3x^2 + x
---------
x + 3

Next, we multiply the divisor ($x + 3$) by the quotient ($x^2$) and subtract the result from the dividend.

x^3 + 3x^2 + x
---------
x + 3
x^2(x + 3) = x^3 + 3x^2
---------
0 + x

We can see that the result is $x$, which is the next term in the quotient.

Step 2: Continue the Division Process

We continue the division process by dividing the leading term of the dividend ($x$) by the leading term of the divisor ($x$), which gives us $1$.

x^3 + 3x^2 + x
---------
x + 3
x^2(x + 3) = x^3 + 3x^2
---------
0 + x
x(1) = x
---------
0 + 3

We can see that the result is $3$, which is the remainder.

Step 3: Subtract the Result

The problem asks us to subtract $x^3 + 2x^2$ from the result. We can do this by subtracting the term $x^3 + 2x^2$ from the dividend.

x^3 + 3x^2 + x
---------
x^3 + 2x^2
---------
x + x

We can see that the result is $2x$.

Step 4: Continue the Division Process

We continue the division process by dividing the leading term of the dividend ($2x$) by the leading term of the divisor ($x$), which gives us $2$.

x^3 + 3x^2 + x
---------
x^3 + 2x^2
---------
x + x
2x(1) = 2x
---------
0 + 3

We can see that the result is $3$, which is the remainder.

Conclusion

In this article, we rewrote the given division problem and provided a clear and organized solution. We broke down the problem into smaller steps and used the long division method to divide the polynomial $x^3 + 3x^2 + x$ by $x + 3$. We also subtracted $x^3 + 2x^2$ from the result and continued the division process to find the remainder. By following these steps, we can solve polynomial division problems with ease.

Final Answer

The final answer is $\boxed{x^2 + 1}$.

Additional Resources

For more information on polynomial division, please refer to the following resources:

• Khan Academy: Polynomial Division
• Mathway: Polynomial Division
• Wolfram Alpha: Polynomial Division

FAQs

Q: What is polynomial division? A: Polynomial division is a method of dividing one polynomial by another.

Q: How do I perform polynomial division? A: To perform polynomial division, you can use the long division method.

Q: What is the remainder in polynomial division? A: The remainder in polynomial division is the amount left over after dividing the dividend by the divisor.

Q: How do I subtract a polynomial from another polynomial? A: To subtract a polynomial from another polynomial, you can subtract the corresponding terms.

Glossary

• Dividend: The polynomial being divided.
• Divisor: The polynomial by which we are dividing.
• Quotient: The result of the division.
• Remainder: The amount left over after dividing the dividend by the divisor.
• Polynomial: An expression consisting of variables and coefficients combined using algebraic operations.
  Polynomial Division Q&A: Frequently Asked Questions =====================================================

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial skill to master, especially when working with complex equations and functions. In this article, we will provide answers to frequently asked questions about polynomial division.

Q&A

Q: What is polynomial division?

A: Polynomial division is a method of dividing one polynomial by another. It is used to simplify complex expressions and solve equations.

Q: How do I perform polynomial division?

A: To perform polynomial division, you can use the long division method. This involves dividing the leading term of the dividend by the leading term of the divisor, and then multiplying the divisor by the quotient and subtracting the result from the dividend.

Q: What is the remainder in polynomial division?

A: The remainder in polynomial division is the amount left over after dividing the dividend by the divisor. It is the part of the dividend that cannot be divided evenly by the divisor.

Q: How do I subtract a polynomial from another polynomial?

A: To subtract a polynomial from another polynomial, you can subtract the corresponding terms. For example, if you have the polynomials $x^2 + 3x + 2$ and $x^2 + 2x + 1$, you can subtract them by subtracting the corresponding terms: $(x^2 + 3x + 2) - (x^2 + 2x + 1) = x + 1$.

Q: What is the difference between polynomial division and long division?

A: Polynomial division and long division are similar, but they are used for different types of numbers. Long division is used for integers, while polynomial division is used for polynomials.

Q: Can I use polynomial division to divide a polynomial by a constant?

A: Yes, you can use polynomial division to divide a polynomial by a constant. In this case, the constant is treated as a polynomial with a degree of 0.

Q: How do I know if a polynomial can be divided by another polynomial?

A: A polynomial can be divided by another polynomial if the degree of the divisor is less than or equal to the degree of the dividend.

Q: What is the quotient in polynomial division?

A: The quotient in polynomial division is the result of the division. It is the polynomial that is obtained by dividing the dividend by the divisor.

Q: Can I use polynomial division to divide a polynomial by a polynomial with a degree of 0?

A: Yes, you can use polynomial division to divide a polynomial by a polynomial with a degree of 0. In this case, the polynomial with a degree of 0 is treated as a constant.

Q: How do I check my work when performing polynomial division?

A: To check your work when performing polynomial division, you can multiply the divisor by the quotient and add the remainder. If the result is equal to the dividend, then your work is correct.

Conclusion

In this article, we provided answers to frequently asked questions about polynomial division. We covered topics such as the definition of polynomial division, how to perform polynomial division, and how to check your work. By mastering polynomial division, you can simplify complex expressions and solve equations with ease.

Additional Resources

For more information on polynomial division, please refer to the following resources:

• Khan Academy: Polynomial Division
• Mathway: Polynomial Division
• Wolfram Alpha: Polynomial Division

Glossary

• Dividend: The polynomial being divided.
• Divisor: The polynomial by which we are dividing.
• Quotient: The result of the division.
• Remainder: The amount left over after dividing the dividend by the divisor.
• Polynomial: An expression consisting of variables and coefficients combined using algebraic operations.
• Degree: The highest power of the variable in a polynomial.
• Constant: A number that does not change value.
• Term: A single part of a polynomial, such as $x^2$ or $3x$.