Using Synthetic Division to Factorize the Volume Expression

Oct 13, 2025 by magicalceylontours.com 48 views

Introduction

In this article, we will explore how to use synthetic division to find the expression for the area of the base of a rectangular prism. The prism has a height of $x+4$ and a volume of $x^3+2x^2-17x-36$ . We will use synthetic division to factorize the volume expression and then find the area of the base.

Understanding the Problem

To find the area of the base of the rectangular prism, we need to first understand the relationship between the volume and the dimensions of the prism. The volume of a rectangular prism is given by the formula $V = lwh$ , where $l$ is the length, $w$ is the width, and $h$ is the height. In this case, the volume is given as $x^3+2x^2-17x-36$ , and the height is given as $x+4$ . We need to find the expression for the area of the base, which is given by the formula $A = lw$ .

Using Synthetic Division to Factorize the Volume Expression

To find the expression for the area of the base, we need to first factorize the volume expression using synthetic division. Synthetic division is a method of dividing a polynomial by a linear factor. In this case, we will divide the volume expression by the linear factor $x+4$ . The volume expression is $x^3+2x^2-17x-36$ , and the linear factor is $x+4$ .

Step 1: Set up the Synthetic Division Table

To perform synthetic division, we need to set up a table with the coefficients of the polynomial and the linear factor. The coefficients of the polynomial are $1, 2, -17, -36$ , and the linear factor is $x+4$ . We will write the coefficients in the table and then perform the division.

	1	2	-17	-36
x+4	1	2	-17	-36

Step 2: Perform the Synthetic Division

To perform the synthetic division, we will multiply the linear factor by the first coefficient and write the result below the second coefficient. We will then add the second coefficient and the result to get the next coefficient. We will repeat this process until we have divided all the coefficients.

	1	2	-17	-36
x+4	1	2	-17	-36
		4	2	4
		6	-15	-32

Step 3: Write the Result

After performing the synthetic division, we will write the result in the form of a polynomial. The result is $x^2+6x-15$ .

Finding the Expression for the Area of the Base

Now that we have factorized the volume expression using synthetic division, we can find the expression for the area of the base. The area of the base is given by the formula $A = lw$ . We can substitute the expression for the volume and the height to get the expression for the area of the base.

Step 1: Substitute the Expression for the Volume and the Height

We will substitute the expression for the volume $x^2+6x-15$ and the height $x+4$ into the formula for the area of the base.

Step 2: Simplify the Expression

We will simplify the expression to get the final result.

Conclusion

In this article, we used synthetic division to find the expression for the area of the base of a rectangular prism with height $x+4$ and volume $x^3+2x^2-17x-36$ . We factorized the volume expression using synthetic division and then found the expression for the area of the base. The final result is $2x^2-9x$ .

Answer

The final answer is A. $2x^2-9x$ .

Introduction

In our previous article, we explored how to use synthetic division to find the expression for the area of the base of a rectangular prism. In this article, we will answer some frequently asked questions (FAQs) about the topic.

Q: What is synthetic division?

A: Synthetic division is a method of dividing a polynomial by a linear factor. It is a shortcut for long division of polynomials and is used to find the roots of a polynomial.

Q: How do I perform synthetic division?

A: To perform synthetic division, you need to set up a table with the coefficients of the polynomial and the linear factor. You will then multiply the linear factor by the first coefficient and write the result below the second coefficient. You will then add the second coefficient and the result to get the next coefficient. You will repeat this process until you have divided all the coefficients.

Q: What is the formula for the area of the base of a rectangular prism?

A: The formula for the area of the base of a rectangular prism is $A = lw$ , where $l$ is the length, $w$ is the width, and $h$ is the height.

Q: How do I find the expression for the area of the base of a rectangular prism using synthetic division?

A: To find the expression for the area of the base of a rectangular prism using synthetic division, you need to first factorize the volume expression using synthetic division. You will then substitute the expression for the volume and the height into the formula for the area of the base and simplify the expression.

Q: What is the relationship between the volume and the dimensions of a rectangular prism?

A: The volume of a rectangular prism is given by the formula $V = lwh$ , where $l$ is the length, $w$ is the width, and $h$ is the height.

Q: Can I use synthetic division to find the expression for the area of the base of a rectangular prism with a non-linear height?

A: No, synthetic division can only be used to find the expression for the area of the base of a rectangular prism with a linear height.

Q: How do I know if the height of the rectangular prism is linear or non-linear?

A: You can determine if the height of the rectangular prism is linear or non-linear by looking at the expression for the height. If the expression is a linear expression, such as $x+4$ , then the height is linear. If the expression is a non-linear expression, such as $x^2+4x+4$ , then the height is non-linear.

Q: Can I use synthetic division to find the expression for the area of the base of a rectangular prism with a non-integer height?

A: Yes, synthetic division can be used to find the expression for the area of the base of a rectangular prism with a non-integer height.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about finding the expression for the area of the base of a rectangular prism using synthetic division. We hope that this article has been helpful in clarifying any confusion about the topic.

Additional Resources

If you are interested in learning more about synthetic division and finding the expression for the area of the base of a rectangular prism, we recommend checking out the following resources:

Textbooks: "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by Michael Spivak
Online Resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha
Software: Mathematica, Maple, and MATLAB

Final Answer

The final answer is A. $2x^2-9x$ .