Divide The Polynomial $x^4 - 3x^2 - 10x + 6$ By $x^2 + 2x + 3$.

$$$$

Introduction

Polynomial division is a fundamental concept in algebra, and it plays a crucial role in solving polynomial equations. In this article, we will learn how to divide the polynomial $x^4 - 3x^2 - 10x + 6$ by $x^2 + 2x + 3$. We will use the long division method to perform this division.

The Long Division Method

The long division method is a step-by-step process that involves dividing the polynomial by the divisor. The process involves the following steps:

1. Divide the leading term of the dividend by the leading term of the divisor.
2. Multiply the entire divisor by the result obtained in step 1.
3. Subtract the product obtained in step 2 from the dividend.
4. Bring down the next term of the dividend.
5. Repeat steps 1-4 until the degree of the remainder is less than the degree of the divisor.

Divide the Polynomial

To divide the polynomial $x^4 - 3x^2 - 10x + 6$ by $x^2 + 2x + 3$, we will use the long division method.

Step 1: Divide the Leading Term

The leading term of the dividend is $x^4$, and the leading term of the divisor is $x^2$. We will divide $x^4$ by $x^2$ to obtain $x^2$.

Step 2: Multiply the Divisor

We will multiply the entire divisor $x^2 + 2x + 3$ by $x^2$ to obtain $x^4 + 2x^3 + 3x^2$.

Step 3: Subtract the Product

We will subtract the product obtained in step 2 from the dividend $x^4 - 3x^2 - 10x + 6$ to obtain $-2x^3 - 3x^2 - 10x + 6$.

Step 4: Bring Down the Next Term

We will bring down the next term of the dividend, which is $-10x$.

Step 5: Repeat the Process

We will repeat the process by dividing the leading term of the new dividend $-2x^3 - 3x^2 - 10x + 6$ by the leading term of the divisor $x^2 + 2x + 3$. We will obtain $-2x$.

Step 6: Multiply the Divisor

We will multiply the entire divisor $x^2 + 2x + 3$ by $-2x$ to obtain $-2x^3 - 4x^2 - 6x$.

Step 7: Subtract the Product

We will subtract the product obtained in step 6 from the new dividend $-2x^3 - 3x^2 - 10x + 6$ to obtain $x^2 - 4x + 6$.

Step 8: Bring Down the Next Term

We will bring down the next term of the new dividend, which is $6$.

Step 9: Repeat the Process

We will repeat the process by dividing the leading term of the new dividend $x^2 - 4x + 6$ by the leading term of the divisor $x^2 + 2x + 3$. We will obtain $1$.

Step 10: Multiply the Divisor

We will multiply the entire divisor $x^2 + 2x + 3$ by $1$ to obtain $x^2 + 2x + 3$.

Step 11: Subtract the Product

We will subtract the product obtained in step 10 from the new dividend $x^2 - 4x + 6$ to obtain $-6x + 3$.

Step 12: Bring Down the Next Term

We will bring down the next term of the new dividend, which is $0$.

Step 13: Repeat the Process

We will repeat the process by dividing the leading term of the new dividend $-6x + 3$ by the leading term of the divisor $x^2 + 2x + 3$. We will obtain $0$.

Step 14: Obtain the Quotient and Remainder

We will obtain the quotient $x^2 - 2x + 1$ and the remainder $-6x + 3$.

Conclusion

In this article, we learned how to divide the polynomial $x^4 - 3x^2 - 10x + 6$ by $x^2 + 2x + 3$ using the long division method. We obtained the quotient $x^2 - 2x + 1$ and the remainder $-6x + 3$.

Final Answer

The final answer is $\boxed{x^2 - 2x + 1}$ with a remainder of $\boxed{-6x + 3}$.

$$$$

Introduction

In our previous article, we learned how to divide the polynomial $x^4 - 3x^2 - 10x + 6$ by $x^2 + 2x + 3$ using the long division method. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the quotient of the division?

A: The quotient of the division is $x^2 - 2x + 1$.

Q: What is the remainder of the division?

A: The remainder of the division is $-6x + 3$.

Q: Why do we need to perform polynomial division?

A: Polynomial division is a fundamental concept in algebra, and it plays a crucial role in solving polynomial equations. It is used to simplify complex expressions and to find the roots of polynomial equations.

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

A: Polynomial division and long division are two different methods of dividing polynomials. Long division is a step-by-step process that involves dividing the polynomial by the divisor, while polynomial division is a more general method that involves dividing the polynomial by the divisor and finding the quotient and remainder.

Q: How do we know when to stop performing polynomial division?

A: We know when to stop performing polynomial division when the degree of the remainder is less than the degree of the divisor.

Q: Can we use polynomial division to divide a polynomial by a non-polynomial expression?

A: No, we cannot use polynomial division to divide a polynomial by a non-polynomial expression. Polynomial division is only used to divide polynomials by other polynomials.

Q: What is the importance of polynomial division in real-world applications?

A: Polynomial division is an important concept in many real-world applications, including engineering, physics, and computer science. It is used to model and analyze complex systems, and to find the roots of polynomial equations.

Conclusion

In this article, we answered some frequently asked questions related to polynomial division. We hope that this article has provided you with a better understanding of this important concept.

Final Answer

The final answer is $\boxed{x^2 - 2x + 1}$ with a remainder of $\boxed{-6x + 3}$.

Additional Resources

• Polynomial Division Tutorial
• Long Division Tutorial
• Polynomial Division Examples

Related Articles

• Divide the Polynomial $x^3 - 2x^2 - 5x + 6$ by $x^2 + 3x + 2$
• Divide the Polynomial $x^4 - 4x^3 - 3x^2 + 2x + 1$ by $x^2 + 2x + 1$
• Divide the Polynomial $x^5 - 3x^4 - 2x^3 + 4x^2 - 5x + 1$ by $x^3 + 2x^2 + 3x + 1$