Select The Correct Answer.Using Long Division, What Is The Quotient Of $3x^4 + 20x^3 + 14x^2 + 17x + 30$ And $x + 6$?A. $2x^3 + 14x^2 + 17x + 30$ B. $ 3 X 3 − 2 X 2 + 2 X − 5 3x^3 - 2x^2 + 2x - 5 3 X 3 − 2 X 2 + 2 X − 5 [/tex] C. $3x^3 + 2x^2 +
Understanding the Problem
When performing long division of polynomials, we are given two polynomials and asked to find the quotient. In this case, we are given the dividend $3x^4 + 20x^3 + 14x^2 + 17x + 30$ and the divisor $x + 6$. Our goal is to find the quotient of these two polynomials.
The Process of Long Division
To perform long division of polynomials, we follow a similar process to long division of numbers. We start by dividing the leading term of the dividend by the leading term of the divisor. In this case, we divide $3x^4$ by $x$, which gives us $3x^3$. We then multiply the entire divisor by $3x^3$ and subtract the result from the dividend.
Step 1: Divide the Leading Term
We start by dividing the leading term of the dividend, $3x^4$, by the leading term of the divisor, $x$. This gives us $3x^3$.
Step 2: Multiply the Divisor
We then multiply the entire divisor, $x + 6$, by the result from Step 1, $3x^3$. This gives us $3x^4 + 18x^3$.
Step 3: Subtract the Result
We subtract the result from Step 2, $3x^4 + 18x^3$, from the dividend, $3x^4 + 20x^3 + 14x^2 + 17x + 30$. This gives us $2x^3 + 14x^2 + 17x + 30$.
Step 4: Repeat the Process
We repeat the process by dividing the leading term of the result from Step 3, $2x^3$, by the leading term of the divisor, $x$. This gives us $2x^2$. We then multiply the entire divisor by $2x^2$ and subtract the result from the result from Step 3.
Step 5: Continue the Process
We continue the process by dividing the leading term of the result from Step 4, $2x^2$, by the leading term of the divisor, $x$. This gives us $2x$. We then multiply the entire divisor by $2x$ and subtract the result from the result from Step 4.
Step 6: Final Result
After repeating the process several times, we arrive at the final result, which is the quotient of the two polynomials.
The Correct Answer
The correct answer is B. $3x^3 - 2x^2 + 2x - 5$.
Explanation
To arrive at the correct answer, we need to perform the long division of polynomials. We start by dividing the leading term of the dividend by the leading term of the divisor, which gives us $3x^3$. We then multiply the entire divisor by $3x^3$ and subtract the result from the dividend. We repeat the process several times until we arrive at the final result.
Conclusion
In conclusion, the correct answer is B. $3x^3 - 2x^2 + 2x - 5$. This is the result of performing the long division of polynomials using the given dividend and divisor.
Example
Here is an example of how to perform the long division of polynomials:
Final Answer
The final answer is B. $3x^3 - 2x^2 + 2x - 5$.
Understanding the Basics
Long division of polynomials is a process used to divide one polynomial by another. It is similar to long division of numbers, but with polynomials. In this article, we will answer some common questions about long division of polynomials.
Q: What is the purpose of long division of polynomials?
A: The purpose of long division of polynomials is to divide one polynomial by another and find the quotient. This is useful in algebra and other areas of mathematics.
Q: How do I perform long division of polynomials?
A: To perform long division of polynomials, you need to follow these steps:
- Divide the leading term of the dividend by the leading term of the divisor.
- Multiply the entire divisor by the result from step 1.
- Subtract the result from step 2 from the dividend.
- Repeat steps 1-3 until you arrive at the final result.
Q: What is the difference between long division of polynomials and long division of numbers?
A: The main difference between long division of polynomials and long division of numbers is that polynomials have variables, while numbers do not. This means that you need to consider the variables when performing long division of polynomials.
Q: How do I handle variables with exponents when performing long division of polynomials?
A: When performing long division of polynomials, you need to handle variables with exponents carefully. You can do this by following these steps:
- Divide the leading term of the dividend by the leading term of the divisor.
- Multiply the entire divisor by the result from step 1.
- Subtract the result from step 2 from the dividend.
- Repeat steps 1-3 until you arrive at the final result.
Q: What is the quotient of the polynomial $3x^4 + 20x^3 + 14x^2 + 17x + 30$ and the polynomial $x + 6$?
A: The quotient of the polynomial $3x^4 + 20x^3 + 14x^2 + 17x + 30$ and the polynomial $x + 6$ is $3x^3 - 2x^2 + 2x - 5$.
Q: How do I check my work when performing long division of polynomials?
A: To check your work when performing long division of polynomials, you can follow these steps:
- Multiply the entire divisor by the result from step 1.
- Subtract the result from step 2 from the dividend.
- Repeat steps 1-2 until you arrive at the final result.
- Check that the final result is correct by multiplying the entire divisor by the result from step 1 and subtracting the result from step 2 from the dividend.
Q: What are some common mistakes to avoid when performing long division of polynomials?
A: Some common mistakes to avoid when performing long division of polynomials include:
- Not handling variables with exponents carefully
- Not following the correct order of operations
- Not checking your work carefully
Conclusion
In conclusion, long division of polynomials is a process used to divide one polynomial by another. It is similar to long division of numbers, but with polynomials. By following the steps outlined in this article, you can perform long division of polynomials and find the quotient.
Example
Here is an example of how to perform long division of polynomials:
Final Answer
The final answer is B. $3x^3 - 2x^2 + 2x - 5$.