What Is The Quotient Of ( 3 X 4 − 4 X 2 + 8 X − 1 ) ÷ ( X − 2 ) ? \left(3 X^4-4 X^2+8 X-1\right) \div (x-2)? ( 3 X 4 − 4 X 2 + 8 X − 1 ) ÷ ( X − 2 )? A. 3 X 3 + 6 X 2 + 8 X + 24 − 47 X − 2 3 X^3 + 6 X^2 + 8 X + 24 - \frac{47}{x-2} 3 X 3 + 6 X 2 + 8 X + 24 − X − 2 47 ​ B. 3 X 3 + 6 X 2 + 8 X + 24 + 47 3 X 4 − 4 X 2 + 8 X − 1 3 X^3 + 6 X^2 + 8 X + 24 + \frac{47}{3 X^4 - 4 X^2 + 8 X - 1} 3 X 3 + 6 X 2 + 8 X + 24 + 3 X 4 − 4 X 2 + 8 X − 1 47 ​ C. $3 X^3 + 6 X^2 + 8 X + 24 +

$$

Introduction

When it comes to polynomial division, there are several steps involved in finding the quotient and remainder. In this article, we will focus on finding the quotient of the given polynomial division problem: $\left(3 x^4-4 x^2+8 x-1\right) \div (x-2)$. We will use the long division method to solve this problem and provide the correct answer.

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.

Long Division Method

The long division method is a step-by-step process of dividing a polynomial by another polynomial. It involves the following 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. Bring down the next term of the dividend and repeat the process.
4. Continue this process until the degree of the remainder is less than the degree of the divisor.

Solving the Problem

To solve the problem, we will use the long division method. We will divide the polynomial $3 x^4-4 x^2+8 x-1$ by the polynomial $x-2$.

Step 1: Divide the Highest Degree Term

The highest degree term of the dividend is $3 x^4$ and the highest degree term of the divisor is $x$. Therefore, we will divide $3 x^4$ by $x$ to get $3 x^3$.

Step 2: Multiply the Divisor by the Result

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

Step 3: Subtract the Result from the Dividend

We will subtract the result $3 x^4 - 6 x^3$ from the dividend $3 x^4-4 x^2+8 x-1$ to get $6 x^3 - 4 x^2 + 8 x - 1$.

Step 4: Bring Down the Next Term

We will bring down the next term of the dividend, which is $-4 x^2$.

Step 5: Repeat the Process

We will repeat the process by dividing the highest degree term of the new dividend $6 x^3 - 4 x^2 + 8 x - 1$ by the highest degree term of the divisor $x$. We will get $6 x^2$.

Step 6: Multiply the Divisor by the Result

We will multiply the entire divisor $x-2$ by the result $6 x^2$ to get $6 x^3 - 12 x^2$.

Step 7: Subtract the Result from the Dividend

We will subtract the result $6 x^3 - 12 x^2$ from the dividend $6 x^3 - 4 x^2 + 8 x - 1$ to get $8 x^2 + 8 x - 1$.

Step 8: Bring Down the Next Term

We will bring down the next term of the dividend, which is $8 x$.

Step 9: Repeat the Process

We will repeat the process by dividing the highest degree term of the new dividend $8 x^2 + 8 x - 1$ by the highest degree term of the divisor $x$. We will get $8 x$.

Step 10: Multiply the Divisor by the Result

We will multiply the entire divisor $x-2$ by the result $8 x$ to get $8 x^2 - 16 x$.

Step 11: Subtract the Result from the Dividend

We will subtract the result $8 x^2 - 16 x$ from the dividend $8 x^2 + 8 x - 1$ to get $24 x - 1$.

Step 12: Bring Down the Next Term

We will bring down the next term of the dividend, which is $-1$.

Step 13: Repeat the Process

We will repeat the process by dividing the highest degree term of the new dividend $24 x - 1$ by the highest degree term of the divisor $x$. We will get $24$.

Step 14: Multiply the Divisor by the Result

We will multiply the entire divisor $x-2$ by the result $24$ to get $24 x - 48$.

Step 15: Subtract the Result from the Dividend

We will subtract the result $24 x - 48$ from the dividend $24 x - 1$ to get $47$.

Conclusion

The final result of the polynomial division is $3 x^3 + 6 x^2 + 8 x + 24 + \frac{47}{x-2}$. This is the correct answer to the problem.

Discussion

The quotient of the polynomial division is $3 x^3 + 6 x^2 + 8 x + 24$. The remainder is $\frac{47}{x-2}$. This result can be used to solve various problems in mathematics and engineering.

Final Answer

The final answer is $3 x^3 + 6 x^2 + 8 x + 24 + \frac{47}{x-2}$.

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing a polynomial by another polynomial. In the previous article, we solved the problem of finding the quotient of $\left(3 x^4-4 x^2+8 x-1\right) \div (x-2)$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on 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: What is the long division method?

A: The long division method is a step-by-step process of dividing a polynomial by another polynomial. It involves the following 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. Bring down the next term of the dividend and repeat the process.
4. Continue this process until the degree of the remainder is less than the degree of the divisor.

Q: How do I know when to stop the long division process?

A: You will know when to stop the long division process when the degree of the remainder is less than the degree of the divisor. At this point, you can write the remainder as a fraction and the quotient as a polynomial.

Q: What is the remainder in polynomial division?

A: The remainder in polynomial division is the amount left over after the division process is complete. It is usually written as a fraction and is denoted by the symbol "R".

Q: How do I write the remainder as a fraction?

A: To write the remainder as a fraction, you will need to divide the remainder by the divisor. This will give you a fraction that represents the remainder.

Q: What is the quotient in polynomial division?

A: The quotient in polynomial division is the result of the division process. It is usually written as a polynomial and is denoted by the symbol "Q".

Q: How do I write the quotient as a polynomial?

A: To write the quotient as a polynomial, you will need to multiply the divisor by the result of each division step. This will give you a polynomial that represents the quotient.

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
• Not multiplying the entire divisor by the result
• Not subtracting the result from the dividend
• Not bringing down the next term of the dividend
• Not repeating the process until the degree of the remainder is less than the degree of the divisor

Q: How can I practice polynomial division?

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

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

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

• Engineering: Polynomial division is used to solve problems in engineering, such as designing electrical circuits and mechanical systems.
• Computer Science: Polynomial division is used in computer science to solve problems in algorithms and data structures.
• Physics: Polynomial division is used in physics to solve problems in mechanics and electromagnetism.

Conclusion

Polynomial division is a fundamental concept in algebra that involves dividing a polynomial by another polynomial. In this article, we provided a Q&A section to help clarify any doubts and provide additional information on polynomial division. We hope this article has been helpful in understanding polynomial division and its applications.