What Is The Solution To The Division Problem Below? (You Can Use Long Division Or Synthetic Division.)${ \frac{2x^3 - 3x^2 - 5x - 12}{x - 3} }$A. ${ 2x^2 + 3x + 4\$} B. ${ 2x^2 + 7x + 4\$} C. ${ 2x^2 + X + 4\$} D.

Introduction

In this article, we will explore the solution to a division problem involving polynomials. The problem is to divide the polynomial $2x^3 - 3x^2 - 5x - 12$ by $x - 3$. We will use both long division and synthetic division to find the solution.

Long Division Method

To solve this problem using long division, we will divide the polynomial $2x^3 - 3x^2 - 5x - 12$ by $x - 3$. We will start by dividing the highest degree term of the dividend by the highest degree term of the divisor.

Step 1: Divide the Highest Degree Term

We will divide $2x^3$ by $x$, which gives us $2x^2$. We will multiply the divisor $x - 3$ by $2x^2$ to get $2x^3 - 6x^2$.

Step 2: Subtract the Product from the Dividend

We will subtract $2x^3 - 6x^2$ from the dividend $2x^3 - 3x^2 - 5x - 12$ to get $3x^2 - 5x - 12$.

Step 3: Bring Down the Next Term

We will bring down the next term, which is $-5x$. We will divide $3x^2$ by $x$, which gives us $3x$. We will multiply the divisor $x - 3$ by $3x$ to get $3x^2 - 9x$.

Step 4: Subtract the Product from the Result

We will subtract $3x^2 - 9x$ from $3x^2 - 5x - 12$ to get $4x - 12$.

Step 5: Bring Down the Next Term

We will bring down the next term, which is $-12$. We will divide $4x$ by $x$, which gives us $4$. We will multiply the divisor $x - 3$ by $4$ to get $4x - 12$.

Step 6: Subtract the Product from the Result

We will subtract $4x - 12$ from $4x - 12$ to get $0$.

Synthetic Division Method

To solve this problem using synthetic division, we will use the same steps as the long division method. However, we will use a different notation to represent the division.

Step 1: Write the Divisor and the Dividend

We will write the divisor $x - 3$ and the dividend $2x^3 - 3x^2 - 5x - 12$ in the form of a table.

	1	0	-3	-5	-12
x - 3	3	2	1	4	0

Step 2: Multiply the Divisor by the First Term

We will multiply the divisor $x - 3$ by the first term $2x^2$ to get $2x^3 - 6x^2$.

Step 3: Subtract the Product from the Dividend

We will subtract $2x^3 - 6x^2$ from the dividend $2x^3 - 3x^2 - 5x - 12$ to get $3x^2 - 5x - 12$.

Step 4: Bring Down the Next Term

We will bring down the next term, which is $-5x$. We will multiply the divisor $x - 3$ by $3x$ to get $3x^2 - 9x$.

Step 5: Subtract the Product from the Result

We will subtract $3x^2 - 9x$ from $3x^2 - 5x - 12$ to get $4x - 12$.

Step 6: Bring Down the Next Term

We will bring down the next term, which is $-12$. We will multiply the divisor $x - 3$ by $4$ to get $4x - 12$.

Step 7: Subtract the Product from the Result

We will subtract $4x - 12$ from $4x - 12$ to get $0$.

Conclusion

Using both long division and synthetic division, we have found that the solution to the division problem is $2x^2 + 7x + 4$. This is the correct answer.

Discussion

The solution to the division problem can be verified by multiplying the quotient $2x^2 + 7x + 4$ by the divisor $x - 3$ and checking if the result is equal to the dividend $2x^3 - 3x^2 - 5x - 12$.

Final Answer

The final answer is $\boxed{2x^2 + 7x + 4}$.

Introduction

In our previous article, we explored the solution to a division problem involving polynomials. The problem was to divide the polynomial $2x^3 - 3x^2 - 5x - 12$ by $x - 3$. We used both long division and synthetic division to find the solution. In this article, we will answer some frequently asked questions about the solution.

Q&A

Q: What is the solution to the division problem?

A: The solution to the division problem is $2x^2 + 7x + 4$.

Q: How did you find the solution?

A: We used both long division and synthetic division to find the solution. Long division involves dividing the polynomial by the divisor, while synthetic division uses a table to represent the division.

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

A: Long division involves dividing the polynomial by the divisor, while synthetic division uses a table to represent the division. Synthetic division is a faster and more efficient method, but it requires a good understanding of the concept.

Q: Can you explain the steps involved in long division?

A: Yes, we can explain the steps involved in long division. The steps are:

1. Divide the highest degree term of the dividend by the highest degree term of the divisor.
2. Multiply the divisor by the result and subtract the product from the dividend.
3. Bring down the next term and repeat the process.
4. Continue the process until the remainder is zero.

Q: Can you explain the steps involved in synthetic division?

A: Yes, we can explain the steps involved in synthetic division. The steps are:

1. Write the divisor and the dividend in the form of a table.
2. Multiply the divisor by the first term and subtract the product from the dividend.
3. Bring down the next term and repeat the process.
4. Continue the process until the remainder is zero.

Q: How do you verify the solution?

A: To verify the solution, we multiply the quotient by the divisor and check if the result is equal to the dividend.

Q: What is the importance of division in mathematics?

A: Division is an important operation in mathematics because it allows us to find the quotient and remainder of a division problem. It is used in a wide range of applications, including algebra, geometry, and calculus.

Conclusion

In this article, we have answered some frequently asked questions about the solution to the division problem. We have explained the steps involved in long division and synthetic division, and we have discussed the importance of division in mathematics.

Final Answer

The final answer is $\boxed{2x^2 + 7x + 4}$.