Study The Partial Work Shown For The Following Division Problem:Divide 6 X 3 + 1 − 14 X 6x^3 + 1 - 14x 6 X 3 + 1 − 14 X By 3 X + 6 3x + 6 3 X + 6 .$[ \begin{array}{r} \frac{2x^2}{3x + 6} \begin{array}{|l} \frac{6x^3 - 14x + 1}{-26x^2 + 1} \end{array} \ \frac{6x^3 +

Introduction

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, particularly in solving equations and simplifying expressions. In this article, we will study the partial work shown for the division problem: Divide $6x^3 + 1 - 14x$ by $3x + 6$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Problem

The given problem is to divide the polynomial $6x^3 + 1 - 14x$ by $3x + 6$. To begin, we need to understand the concept of polynomial division. Polynomial division involves dividing a polynomial by another polynomial, resulting in a quotient and a remainder. The quotient is the result of the division, while the remainder is the amount left over after the division.

Step 1: Write the Division Problem

To start the division, we write the dividend (the polynomial being divided) and the divisor (the polynomial by which we are dividing) in the correct format. The dividend is $6x^3 + 1 - 14x$, and the divisor is $3x + 6$. We can write the division problem as follows:

$$\frac{6x^3 + 1 - 14x}{3x + 6}$$

Step 2: Divide the Leading Term

The first step in polynomial division is to divide the leading term of the dividend by the leading term of the divisor. In this case, the leading term of the dividend is $6x^3$, and the leading term of the divisor is $3x$. We can divide $6x^3$ by $3x$ to get $2x^2$. This is the first term of the quotient.

Step 3: Multiply the Divisor by the Quotient

Next, we multiply the divisor ($3x + 6$) by the quotient term ($2x^2$) to get $6x^3 + 12x^2$. We then subtract this result from the dividend ($6x^3 + 1 - 14x$) to get a new polynomial.

Step 4: Bring Down the Next Term

After subtracting $6x^3 + 12x^2$ from the dividend, we bring down the next term, which is $-14x$. The new polynomial is $-14x - 12x^2 + 1$.

Step 5: Repeat the Process

We repeat the process by dividing the leading term of the new polynomial ($-14x$) by the leading term of the divisor ($3x$) to get $-4.67x$. However, since we are working with polynomials, we can only divide by exact values. Therefore, we will divide $-14x$ by $3x$ to get $-4\frac{2}{3}x$. This is the next term of the quotient.

Step 6: Multiply the Divisor by the Quotient

Next, we multiply the divisor ($3x + 6$) by the quotient term ($-4\frac{2}{3}x$) to get $-12x - 24$. We then subtract this result from the new polynomial ($-14x - 12x^2 + 1$) to get a new polynomial.

Step 7: Bring Down the Next Term

After subtracting $-12x - 24$ from the new polynomial, we bring down the next term, which is $1$. The new polynomial is $-2x^2 - 1$.

Step 8: Repeat the Process

We repeat the process by dividing the leading term of the new polynomial ($-2x^2$) by the leading term of the divisor ($3x$) to get $-\frac{2}{3}x$. This is the next term of the quotient.

Step 9: Multiply the Divisor by the Quotient

Next, we multiply the divisor ($3x + 6$) by the quotient term ($-\frac{2}{3}x$) to get $-2x - 4$. We then subtract this result from the new polynomial ($-2x^2 - 1$) to get a new polynomial.

Step 10: Bring Down the Next Term

After subtracting $-2x - 4$ from the new polynomial, we bring down the next term, which is $1$. The new polynomial is $-26x^2 + 1$.

Step 11: Repeat the Process

We repeat the process by dividing the leading term of the new polynomial ($-26x^2$) by the leading term of the divisor ($3x$) to get $-\frac{26}{3}x^2$. However, since we are working with polynomials, we can only divide by exact values. Therefore, we will divide $-26x^2$ by $3x$ to get $-\frac{26}{3}x^2$. This is the next term of the quotient.

Step 12: Multiply the Divisor by the Quotient

Next, we multiply the divisor ($3x + 6$) by the quotient term ($-\frac{26}{3}x^2$) to get $-26x^3 - 52x^2$. We then subtract this result from the new polynomial ($-26x^2 + 1$) to get a new polynomial.

Step 13: Bring Down the Next Term

After subtracting $-26x^3 - 52x^2$ from the new polynomial, we bring down the next term, which is $1$. The new polynomial is $-78x^2 + 1$.

Step 14: Repeat the Process

We repeat the process by dividing the leading term of the new polynomial ($-78x^2$) by the leading term of the divisor ($3x$) to get $-26x$. This is the next term of the quotient.

Step 15: Multiply the Divisor by the Quotient

Next, we multiply the divisor ($3x + 6$) by the quotient term ($-26x$) to get $-78x^2 - 156x$. We then subtract this result from the new polynomial ($-78x^2 + 1$) to get a new polynomial.

Step 16: Bring Down the Next Term

After subtracting $-78x^2 - 156x$ from the new polynomial, we bring down the next term, which is $1$. The new polynomial is $157x + 1$.

Step 17: Repeat the Process

We repeat the process by dividing the leading term of the new polynomial ($157x$) by the leading term of the divisor ($3x$) to get $52\frac{1}{3}x$. However, since we are working with polynomials, we can only divide by exact values. Therefore, we will divide $157x$ by $3x$ to get $52\frac{1}{3}x$. This is the next term of the quotient.

Step 18: Multiply the Divisor by the Quotient

Next, we multiply the divisor ($3x + 6$) by the quotient term ($52\frac{1}{3}x$) to get $156x + 312$. We then subtract this result from the new polynomial ($157x + 1$) to get a new polynomial.

Step 19: Bring Down the Next Term

After subtracting $156x + 312$ from the new polynomial, we bring down the next term, which is $1$. The new polynomial is $x - 311$.

Step 20: Repeat the Process

We repeat the process by dividing the leading term of the new polynomial ($x$) by the leading term of the divisor ($3x$) to get $\frac{1}{3}$. This is the next term of the quotient.

Step 21: Multiply the Divisor by the Quotient

Next, we multiply the divisor ($3x + 6$) by the quotient term ($\frac{1}{3}$) to get $x + 2$. We then subtract this result from the new polynomial ($x - 311$) to get a new polynomial.

Step 22: Bring Down the Next Term

After subtracting $x + 2$ from the new polynomial, we bring down the next term, which is $-311$. The new polynomial is $-312$.

Step 23: Repeat the Process

We repeat the process by dividing the leading term of the new polynomial ($-312$) by the leading term of the divisor ($3x$) to get $-104$. However, since we are working with polynomials, we can only divide by exact values. Therefore, we will divide $-312$ by $3x$ to get $-104$. This is the next term of the quotient.

Step 24: Multiply the Divisor by the Quotient

$$$$$$$$

Q&A: Frequently Asked Questions

Q: What is polynomial division? A: Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, particularly in solving equations and simplifying expressions.

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 mathematics, particularly in algebra and calculus.

Q: How do I divide a polynomial by another polynomial? A: To divide a polynomial by another polynomial, you need to follow the steps outlined in this article. You need to write the division problem, divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by the quotient, and bring down the next term.

Q: What is the quotient in polynomial division? A: The quotient in polynomial division is the result of the division. It is the polynomial that results from dividing the dividend by the divisor.

Q: What is the remainder in polynomial division? A: The remainder in polynomial division is the amount left over after the division. It is the polynomial that results from subtracting the product of the divisor and the quotient from the dividend.

Q: How do I determine the degree of the quotient? A: To determine the degree of the quotient, you need to look at the highest power of the variable in the quotient. The degree of the quotient is the highest power of the variable.

Q: How do I determine the degree of the remainder? A: To determine the degree of the remainder, you need to look at the highest power of the variable in the remainder. The degree of the remainder is the highest power of the variable.

Q: What is the difference between polynomial division and long division? A: Polynomial division and long division are similar, but they are used for different types of numbers. Polynomial division is used for polynomials, while long division is used for integers.

Q: Can I use polynomial division to divide a polynomial by a binomial? A: Yes, you can use polynomial division to divide a polynomial by a binomial. The process is similar to dividing a polynomial by a polynomial.

Q: Can I use polynomial division to divide a polynomial by a trinomial? A: Yes, you can use polynomial division to divide a polynomial by a trinomial. The process is similar to dividing a polynomial by a polynomial.

Q: What are some common mistakes to avoid when using polynomial division? A: Some common mistakes to avoid when using polynomial division include:

• Not writing the division problem correctly
• Not dividing the leading term of the dividend by the leading term of the divisor
• Not multiplying the divisor by the quotient
• Not bringing down the next term
• Not checking the degree of the quotient and the remainder

Conclusion

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, particularly in solving equations and simplifying expressions. By following the steps outlined in this article, you can learn how to divide a polynomial by another polynomial and understand the concept of polynomial division.

Additional Resources

• Polynomial Division Tutorial
• Polynomial Division Examples
• Polynomial Division Practice Problems

Glossary

• Polynomial: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• Dividend: The dividend is the polynomial being divided.
• Divisor: The divisor is the polynomial by which we are dividing.
• Quotient: The quotient is the result of the division.
• Remainder: The remainder is the amount left over after the division.
• Degree: The degree of a polynomial is the highest power of the variable in the polynomial.