What Is The First Step Of The Following Division Problem? ( 8 X 3 − X 2 + 6 X + 7 ) ÷ ( 2 X − 1 \left(8x^3 - X^2 + 6x + 7\right) \div (2x - 1 ( 8 X 3 − X 2 + 6 X + 7 ) ÷ ( 2 X − 1 ]A. Divide 8 X 3 8x^3 8 X 3 By 2 X 2x 2 X .B. Divide 2 X 2x 2 X By 8 X 3 8x^3 8 X 3 .C. Divide 6 X 6x 6 X By 2 X 2x 2 X .

Introduction

When it comes to solving division problems involving polynomials, it's essential to understand the correct order of operations. In this article, we will delve into the first step of dividing a polynomial by another polynomial, specifically the division of $\left(8x^3 - x^2 + 6x + 7\right)$ by $(2x - 1)$. We will explore the correct approach to solving this problem and provide a step-by-step guide.

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.

The First Step of Polynomial Division

The first step of polynomial division is to divide the highest degree term of the dividend by the highest degree term of the divisor. In this case, the highest degree term of the dividend is $8x^3$ and the highest degree term of the divisor is $2x$. To divide $8x^3$ by $2x$, we need to divide the coefficients and subtract the exponents.

Dividing Coefficients and Subtracting Exponents

To divide $8x^3$ by $2x$, we need to divide the coefficients $8$ and $2$, and subtract the exponents $3$ and $1$. This can be done as follows:

$$\frac{8x^3}{2x} = \frac{8}{2}x^{3-1} = 4x^2$$

Conclusion

In conclusion, the first step of the division problem $\left(8x^3 - x^2 + 6x + 7\right) \div (2x - 1)$ is to divide $8x^3$ by $2x$. This involves dividing the coefficients and subtracting the exponents, resulting in $4x^2$. This is the correct approach to solving this problem, and it sets the stage for the next steps in the division process.

Step-by-Step Guide

Here is a step-by-step guide to solving the division problem:

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

Example

Let's use an example to illustrate the first step of polynomial division. Suppose we want to divide $12x^2 + 3x + 2$ by $3x + 1$. The first step would be to divide the highest degree term of the dividend, $12x^2$, by the highest degree term of the divisor, $3x$.

$$\frac{12x^2}{3x} = \frac{12}{3}x^{2-1} = 4x$$

Discussion

The first step of polynomial division is a crucial step in solving division problems involving polynomials. 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.

Final Answer

The final answer to the first step of the division problem $\left(8x^3 - x^2 + 6x + 7\right) \div (2x - 1)$ is $4x^2$.

Introduction

Polynomial division is a fundamental concept in algebra that can be challenging to understand, especially for beginners. In this article, we will address some of the most frequently asked questions about polynomial division, providing clear and concise answers to help you better understand this concept.

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 do we need to divide the highest degree term first?

A: We need to divide the highest degree term first because it is the term with the highest power of the variable. Dividing this term first allows us to eliminate the highest power of the variable and make the division process easier.

Q: How do I divide a polynomial by another polynomial?

A: To divide a polynomial by another polynomial, 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 steps 1-2 until the degree of the remainder is less than the degree of the divisor.

Q: What is the remainder in polynomial division?

A: The remainder in polynomial division is the expression that is left after the division process is complete. It is the part of the dividend that cannot be divided by the divisor.

Q: Can the remainder be zero?

A: Yes, the remainder can be zero. This occurs when the dividend is exactly divisible by the divisor, and there is no remainder left.

Q: How do I check if the remainder is zero?

A: To check if the remainder is zero, simply perform the division process and see if the degree of the remainder is less than the degree of the divisor. If it is, then the remainder is zero.

Q: What is the quotient in polynomial division?

A: The quotient in polynomial division is the expression that results from the division process. It is the part of the dividend that is divided by the divisor.

Q: Can the quotient be a polynomial with a negative degree?

A: No, the quotient cannot be a polynomial with a negative degree. The degree of the quotient is always less than or equal to the degree of the divisor.

Q: How do I simplify the quotient?

A: To simplify the quotient, simply combine like terms and eliminate any negative exponents.

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

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

• Dividing the wrong term first
• Not multiplying the entire divisor by the result
• Not subtracting the result from the dividend
• Not simplifying the quotient

Q: How do I practice polynomial division?

A: To practice polynomial division, try dividing simple polynomials by hand, such as dividing a linear polynomial by a constant. You can also use online tools or calculators to practice polynomial division.

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

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

• Solving systems of equations
• Finding the roots of a polynomial
• Analyzing the behavior of a polynomial
• Modeling real-world phenomena

Conclusion

Polynomial division is a fundamental concept in algebra that can be challenging to understand, but with practice and patience, you can master it. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in polynomial division and apply it to real-world problems.