What Is The Quotient Of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right$\]?A. $x+2$B. $x-2$C. $x+10$D. $x+6$

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Introduction

In algebra, polynomial division is a process of dividing a polynomial by another polynomial. It is a crucial concept in mathematics, and it has numerous applications in various fields, including engineering, physics, and computer science. In this article, we will focus on finding the quotient of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right)$.

Understanding Polynomial Division

Polynomial division is a process of dividing a polynomial by another polynomial. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.

Step 1: Divide the Highest Degree Term

To find the quotient of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right)$, we start by dividing the highest degree term of the dividend, which is $x^3$, by the highest degree term of the divisor, which is $x^2$. This gives us $x$.

Step 2: Multiply the Divisor by the Result

Next, we multiply the entire divisor, $\left(x^2+4x+3\right)$, by the result, $x$. This gives us $x^3+4x^2+3x$.

Step 3: Subtract the Result from the Dividend

We then subtract the result, $x^3+4x^2+3x$, from the dividend, $\left(x^3+6x^2+11x+6\right)$. This gives us $2x^2+8x+6$.

Step 4: Repeat the Process

We repeat the process by dividing the highest degree term of the new dividend, $2x^2$, by the highest degree term of the divisor, $x^2$. This gives us $2$.

Step 5: Multiply the Divisor by the Result

Next, we multiply the entire divisor, $\left(x^2+4x+3\right)$, by the result, $2$. This gives us $2x^2+8x+6$.

Step 6: Subtract the Result from the Dividend

We then subtract the result, $2x^2+8x+6$, from the new dividend, $2x^2+8x+6$. This gives us $0$.

Conclusion

Since the remainder is $0$, we can conclude that the quotient of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right)$ is $x+2$.

Final Answer

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

Discussion

The quotient of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right)$ is $x+2$. This can be verified by multiplying the quotient by the divisor and adding the remainder. The result should be equal to the dividend.

Example

Let's multiply the quotient, $x+2$, by the divisor, $\left(x^2+4x+3\right)$. This gives us $x^3+2x^2+4x^2+8x+3x+6$. Simplifying this expression, we get $x^3+6x^2+11x+6$, which is equal to the dividend.

Conclusion

In conclusion, the quotient of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right)$ is $x+2$. This can be verified by multiplying the quotient by the divisor and adding the remainder. The result should be equal to the dividend.

Final Answer

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

Related Topics

• Polynomial division
• Algebra
• Mathematics

References

• [1] "Polynomial Division" by Math Open Reference
• [2] "Algebra" by Khan Academy
• [3] "Mathematics" by Wikipedia

Keywords

• Polynomial division
• Algebra
• Mathematics
• Quotient
• Dividend
• Divisor
• Remainder
  

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing a polynomial by another polynomial. It is a crucial skill to master in mathematics, and it has numerous applications in various fields, including engineering, physics, and computer science. In this article, we will answer some frequently asked questions (FAQs) about polynomial division.

Q: What is polynomial division?

A: Polynomial division is a process of dividing a polynomial by another polynomial. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.

Q: Why is polynomial division important?

A: Polynomial division is important because it allows us to simplify complex expressions and solve equations. It is a crucial tool in algebra and has numerous applications in various fields, including engineering, physics, and computer science.

Q: How do I perform polynomial division?

A: To perform polynomial division, you need to follow these steps:

1. Divide the highest degree term of the dividend by the highest degree term of the divisor.
2. Multiply the entire divisor by the result and subtract it from the dividend.
3. Repeat the process until the degree of the remainder is less than the degree of the divisor.

Q: What is the quotient of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right)$?

A: The quotient of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right)$ is $x+2$.

Q: How do I check my answer?

A: To check your answer, you need to multiply the quotient by the divisor and add the remainder. The result should be equal to the dividend.

Q: What is the remainder of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right)$?

A: The remainder of $\left(x^3+6x^2+11x+6\right) \div \left(x^2+4x+3\right)$ is $0$.

Q: Can I use polynomial division to solve equations?

A: Yes, you can use polynomial division to solve equations. By dividing both sides of the equation by a common factor, you can simplify the equation and solve for the variable.

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

A: Some common mistakes to avoid when performing polynomial division include:

• Not following the correct order of operations
• Not multiplying the entire divisor by the result
• Not subtracting the result from the dividend
• Not repeating the process until the degree of the remainder is less than the degree of the divisor

Q: How do I practice polynomial division?

A: You can practice polynomial division by working through examples and exercises. You can also use online resources and tools to help you practice and improve your skills.

Conclusion

In conclusion, polynomial division is a fundamental concept in algebra that involves dividing a polynomial by another polynomial. It is a crucial skill to master in mathematics, and it has numerous applications in various fields, including engineering, physics, and computer science. By following the steps outlined in this article, you can perform polynomial division and solve equations.

Final Answer

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

Related Topics

• Polynomial division
• Algebra
• Mathematics

References

• [1] "Polynomial Division" by Math Open Reference
• [2] "Algebra" by Khan Academy
• [3] "Mathematics" by Wikipedia

Keywords

• Polynomial division
• Algebra
• Mathematics
• Quotient
• Dividend
• Divisor
• Remainder