Use Long Division To Solve The Following:$\[ \left(2x^3 + 7x^2 + 9x - 20\right) \div (x + 3) \\]

Introduction

Long division is a fundamental concept in algebra that allows us to divide polynomials by other polynomials. In this article, we will use long division to solve the polynomial equation $\left(2x^3 + 7x^2 + 9x - 20\right) \div (x + 3)$. We will break down the process into manageable steps and provide a clear explanation of each step.

What is Long Division in Algebra?

Long division in algebra is similar to long division in arithmetic, but it involves dividing polynomials by other polynomials. The process 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. We then subtract the product from the dividend and repeat the process until we have a remainder.

Step 1: Write the Polynomial Equation

The polynomial equation we want to solve is $\left(2x^3 + 7x^2 + 9x - 20\right) \div (x + 3)$. To begin the long division process, we need to write the polynomial equation in the correct format.

\begin{array}{r}
x^2 + \underline{2x + 3} \\
x + 3 \enclose{longdiv}{2x^3 + 7x^2 + 9x - 20} \\
\end{array}

Step 2: Divide the Highest Degree Term

The highest degree term of the dividend is $2x^3$, and the highest degree term of the divisor is $x$. To divide the highest degree term of the dividend by the highest degree term of the divisor, we need to divide $2x^3$ by $x$, which gives us $2x^2$.

\begin{array}{r}
x^2 + \underline{2x + 3} \\
x + 3 \enclose{longdiv}{2x^3 + 7x^2 + 9x - 20} \\
- (2x^3 + 6x^2) \\
\end{array}

Step 3: Multiply the Divisor by the Result

We now multiply the entire divisor by the result we obtained in the previous step, which is $2x^2$. This gives us $2x^3 + 6x^2$.

\begin{array}{r}
x^2 + \underline{2x + 3} \\
x + 3 \enclose{longdiv}{2x^3 + 7x^2 + 9x - 20} \\
- (2x^3 + 6x^2) \\
\hline
x^2 + 9x - 20 \\
\end{array}

Step 4: Subtract the Product from the Dividend

We now subtract the product we obtained in the previous step from the dividend. This gives us $x^2 + 9x - 20$.

\begin{array}{r}
x^2 + \underline{2x + 3} \\
x + 3 \enclose{longdiv}{2x^3 + 7x^2 + 9x - 20} \\
- (2x^3 + 6x^2) \\
\hline
x^2 + 9x - 20 \\
\end{array}

Step 5: Repeat the Process

We now repeat the process by dividing the highest degree term of the new dividend by the highest degree term of the divisor. The highest degree term of the new dividend is $x^2$, and the highest degree term of the divisor is $x$. To divide the highest degree term of the new dividend by the highest degree term of the divisor, we need to divide $x^2$ by $x$, which gives us $x$.

\begin{array}{r}
x^2 + \underline{2x + 3} \\
x + 3 \enclose{longdiv}{2x^3 + 7x^2 + 9x - 20} \\
- (2x^3 + 6x^2) \\
\hline
x^2 + 9x - 20 \\
- (x^2 + 3x) \\
\hline
6x - 20 \\
\end{array}

Step 6: Write the Final Answer

We now have a remainder of $6x - 20$. Since the remainder is a polynomial of degree less than the divisor, we can write the final answer as $x^2 + 2x + 3 + \frac{6x - 20}{x + 3}$.

Conclusion

In this article, we used long division to solve the polynomial equation $\left(2x^3 + 7x^2 + 9x - 20\right) \div (x + 3)$. We broke down the process into manageable steps and provided a clear explanation of each step. We also wrote the final answer in the correct format.

Final Answer

$$

Introduction

Long division is a fundamental concept in algebra that allows us to divide polynomials by other polynomials. In our previous article, we used long division to solve the polynomial equation $\left(2x^3 + 7x^2 + 9x - 20\right) \div (x + 3)$. In this article, we will provide a Q&A guide to help you understand long division in algebra.

Q: What is long division in algebra?

A: Long division in algebra is similar to long division in arithmetic, but it involves dividing polynomials by other polynomials. The process 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.

Q: How do I start the long division process?

A: To start the long division process, you need to write the polynomial equation in the correct format. This involves writing the dividend and the divisor in the correct format, with the dividend on top and the divisor on the bottom.

Q: What is the first step in the long division process?

A: The first step in the long division process is to divide the highest degree term of the dividend by the highest degree term of the divisor. This will give you the first term of the quotient.

Q: How do I multiply the divisor by the result?

A: To multiply the divisor by the result, you need to multiply the entire divisor by the result you obtained in the previous step. This will give you a product that you can subtract from the dividend.

Q: What is the remainder in long division?

A: The remainder in long division is the amount left over after you have subtracted the product from the dividend. If the remainder is a polynomial of degree less than the divisor, you can write the final answer as the quotient plus the remainder divided by the divisor.

Q: How do I write the final answer?

A: To write the final answer, you need to combine the quotient and the remainder divided by the divisor. This will give you the final answer in the correct format.

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

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

• Not writing the polynomial equation in the correct format
• Not dividing the highest degree term of the dividend by the highest degree term of the divisor
• Not multiplying the divisor by the result
• Not subtracting the product from the dividend
• Not writing the final answer in the correct format

Q: How can I practice long division in algebra?

A: You can practice long division in algebra by working through examples and exercises. You can also use online resources and tools to help you practice and improve your skills.

Conclusion

In this article, we provided a Q&A guide to help you understand long division in algebra. We covered the basics of long division, including how to start the process, how to multiply the divisor by the result, and how to write the final answer. We also discussed common mistakes to avoid and provided tips for practicing long division in algebra.

Final Answer

The final answer is that long division in algebra is a powerful tool for solving polynomial equations. By following the steps outlined in this article, you can master the art of long division and become proficient in solving polynomial equations.

Additional Resources

• Online resources and tools for practicing long division in algebra
• Examples and exercises for practicing long division in algebra
• Video tutorials and lectures on long division in algebra

Common Questions

• What is long division in algebra?
• How do I start the long division process?
• What is the first step in the long division process?
• How do I multiply the divisor by the result?
• What is the remainder in long division?
• How do I write the final answer?
• What are some common mistakes to avoid in long division?
• How can I practice long division in algebra?