Divide:${ X^3 + 43x + 42 \quad \div \quad X + 6 }$A. { X^2 - 37x - 264 $}$ B. { X^2 - 6x - 7 $}$ C. { X^2 + 6x - 7 $}$ D. { X^2 - 49x + 252 $}$

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Introduction

In this article, we will be performing polynomial division, which is a fundamental concept in algebra. Polynomial division is used to divide a polynomial by another polynomial of lower degree. It is an essential tool in mathematics, particularly in algebra and calculus. In this discussion, we will be dividing the polynomial $x^3 + 43x + 42$ by $x + 6$.

The Division Process

To perform polynomial division, we need to follow a series of steps. The first step is to determine the degree of the dividend and the divisor. In this case, the degree of the dividend is 3, and the degree of the divisor is 1. The next step is to divide the leading term of the dividend by the leading term of the divisor. In this case, we divide $x^3$ by $x$, which gives us $x^2$.

Setting Up the Division

Now that we have the first term of the quotient, we need to multiply the divisor by this term and subtract the result from the dividend. The divisor is $x + 6$, and the first term of the quotient is $x^2$. Multiplying the divisor by the first term of the quotient gives us $x^3 + 6x^2$. Subtracting this from the dividend gives us $-6x^2 + 43x + 42$.

Continuing the Division

Now that we have the result of the subtraction, we need to repeat the process. We divide the leading term of the result by the leading term of the divisor. In this case, we divide $-6x^2$ by $x$, which gives us $-6x$. We then multiply the divisor by this term and subtract the result from the dividend.

The Final Result

After repeating the process several times, we arrive at the final result. The quotient is $x^2 - 6x - 7$, and the remainder is 0.

Conclusion

In conclusion, the result of dividing $x^3 + 43x + 42$ by $x + 6$ is $x^2 - 6x - 7$. This is the correct answer, and it can be verified by multiplying the quotient by the divisor and adding the remainder.

Answer Choice Analysis

Let's analyze the answer choices to see which one matches our result.

• A. $x^2 - 37x - 264$: This is not the correct answer, as the coefficient of the $x$ term is not $-6$.
• B. $x^2 - 6x - 7$: This is the correct answer, as it matches our result.
• C. $x^2 + 6x - 7$: This is not the correct answer, as the coefficient of the $x$ term is not $-6$.
• D. $x^2 - 49x + 252$: This is not the correct answer, as the coefficient of the $x$ term is not $-6$.

Final Answer

The final answer is B. $x^2 - 6x - 7$.

Additional Examples

Here are a few additional examples of polynomial division:

• $x^3 + 2x^2 + 3x + 4 \quad \div \quad x + 1$
• $x^3 + 3x^2 + 2x + 1 \quad \div \quad x + 2$
• $x^3 + 4x^2 + 3x + 2 \quad \div \quad x + 3$

These examples demonstrate the process of polynomial division and how to apply it to different polynomials.

Tips and Tricks

Here are a few tips and tricks for performing polynomial division:

• Make sure to follow the order of operations when multiplying and subtracting.
• Use a calculator or computer program to check your work and ensure that the remainder is 0.
• Practice, practice, practice! The more you practice polynomial division, the more comfortable you will become with the process.

Conclusion

In conclusion, polynomial division is a fundamental concept in algebra that is used to divide a polynomial by another polynomial of lower degree. It is an essential tool in mathematics, particularly in algebra and calculus. By following the steps outlined in this article, you can perform polynomial division and arrive at the correct result. Remember to practice, practice, practice, and use a calculator or computer program to check your work and ensure that the remainder is 0.

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Introduction

In our previous article, we discussed the process of polynomial division and how to divide the polynomial $x^3 + 43x + 42$ by $x + 6$. In this article, we will be answering some frequently asked questions about polynomial division.

Q&A

Q: What is polynomial division?

A: Polynomial division is a process of dividing a polynomial by another polynomial of lower degree. It is used to simplify complex polynomials and to find the quotient and remainder of a division.

Q: How do I perform polynomial division?

A: To perform polynomial division, you need to follow a series of steps. The first step is to determine the degree of the dividend and the divisor. The next step is to divide the leading term of the dividend by the leading term of the divisor. You then multiply the divisor by this term and subtract the result from the dividend. You repeat this process until you have a remainder of 0.

Q: What is the quotient and remainder in polynomial division?

A: The quotient is the result of the division, and it is a polynomial of lower degree than the dividend. The remainder is the amount left over after the division, and it is a polynomial of lower degree than the divisor.

Q: How do I check my work in polynomial division?

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

Q: What are some common mistakes to avoid in polynomial division?

A: Some common mistakes to avoid in polynomial division include:

• Not following the order of operations when multiplying and subtracting
• Not checking your work to ensure that the remainder is 0
• Not using a calculator or computer program to check your work

Q: How do I practice polynomial division?

A: To practice polynomial division, you can try dividing different polynomials by each other. You can also use online resources or math textbooks to find practice problems.

Q: What are some real-world applications of polynomial division?

A: Polynomial division has many real-world applications, including:

• Simplifying complex polynomials in engineering and physics
• Finding the roots of a polynomial in computer science and mathematics
• Modeling population growth and decay in biology and economics

Conclusion

In conclusion, polynomial division is a fundamental concept in algebra that is used to divide a polynomial by another polynomial of lower degree. By following the steps outlined in this article, you can perform polynomial division and arrive at the correct result. Remember to practice, practice, practice, and use a calculator or computer program to check your work and ensure that the remainder is 0.

Additional Resources

Here are some additional resources for learning more about polynomial division:

• Online resources: Khan Academy, Mathway, and Wolfram Alpha
• Math textbooks: "Algebra" by Michael Artin and "Calculus" by Michael Spivak
• Online courses: Coursera, edX, and Udemy

Final Answer

The final answer is B. $x^2 - 6x - 7$.

Final Tips

Here are a few final tips for learning more about polynomial division:

• Practice, practice, practice! The more you practice polynomial division, the more comfortable you will become with the process.
• Use online resources and math textbooks to find practice problems and learn more about polynomial division.
• Don't be afraid to ask for help if you are struggling with polynomial division.