Applying Synthetic Division

Oct 13, 2025 by ludoplayers7.com 88 views

Introduction

In this article, we will explore the concept of synthetic division and its application in finding the width of a rectangle given its area and length. Synthetic division is a method used to divide polynomials by linear factors, and it is a powerful tool in algebra and mathematics.

The Problem

The area of a rectangle is given by the polynomial $5x^3 + 19x^2 + 6x - 18$ , and its length is given by the expression $x + 3$ . We are asked to find the width of the rectangle using synthetic division.

Synthetic Division

Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut method that eliminates the need for long division. The process involves dividing the polynomial by a linear factor, which is in the form of $(x - c)$ , where $c$ is a constant.

To perform synthetic division, we need to follow these steps:

Write down the coefficients of the polynomial in a row.
Write down the value of the linear factor, which is $c$ in the expression $(x - c)$ .
Bring down the first coefficient.
Multiply the value of the linear factor by the first coefficient and write the result below the second coefficient.
Add the second coefficient and the result from step 4.
Repeat steps 4 and 5 until you reach the last coefficient.
The final result is the quotient and the remainder.

Applying Synthetic Division

In this problem, we need to divide the polynomial $5x^3 + 19x^2 + 6x - 18$ by the linear factor $(x + 3)$ . We will use synthetic division to find the quotient and the remainder.

1	5	19	6	-18
-3	-5	58	-12	54

The first step is to write down the coefficients of the polynomial in a row. The coefficients are 5, 19, 6, and -18.

The second step is to write down the value of the linear factor, which is -3 in the expression $(x + 3)$ .

The third step is to bring down the first coefficient, which is 5.

The fourth step is to multiply the value of the linear factor by the first coefficient and write the result below the second coefficient. The result is -5.

The fifth step is to add the second coefficient and the result from step 4. The result is 14.

The sixth step is to repeat steps 4 and 5 until you reach the last coefficient.

1	5	19	6	-18
-3	-5	58	-12	54

The final result is the quotient and the remainder. The quotient is $5x^2 + 4x - 6$ , and the remainder is 54.

Finding the Width of the Rectangle

The area of a rectangle is given by the formula $A = l \times w$ , where $A$ is the area, $l$ is the length, and $w$ is the width. We are given the area and the length, and we need to find the width.

The area of the rectangle is given by the polynomial $5x^3 + 19x^2 + 6x - 18$ , and the length is given by the expression $x + 3$ . We can substitute the length into the area formula to get:

$5x^3 + 19x^2 + 6x - 18 = (x + 3) \times w$

We can expand the right-hand side of the equation to get:

$5x^3 + 19x^2 + 6x - 18 = xw + 3w$

We can equate the coefficients of the two polynomials to get:

$5x^3 + 19x^2 + 6x - 18 = xw + 3w$

We can simplify the equation to get:

$5x^3 + 19x^2 + 6x - 18 = (x + 3)w$

We can divide both sides of the equation by $(x + 3)$ to get:

$w = \frac{5x^3 + 19x^2 + 6x - 18}{x + 3}$

We can use synthetic division to divide the polynomial by $(x + 3)$ to get:

$w = 5x^2 + 4x - 6 + \frac{54}{x + 3}$

We can simplify the equation to get:

$w = 5x^2 + 4x - 6 + \frac{54}{x + 3}$

Conclusion

In this article, we used synthetic division to find the width of a rectangle given its area and length. We divided the polynomial $5x^3 + 19x^2 + 6x - 18$ by the linear factor $(x + 3)$ to get the quotient and the remainder. We then used the quotient and the remainder to find the width of the rectangle.

$The final answer is $5x^2 + 4x - 6 + \frac{54}{x + 3}$.<br/> **The Area of a Rectangle: A Synthetic Division Problem - Q&A**$

Introduction

In our previous article, we explored the concept of synthetic division and its application in finding the width of a rectangle given its area and length. We used synthetic division to divide the polynomial $5x^3 + 19x^2 + 6x - 18$ by the linear factor $(x + 3)$ to get the quotient and the remainder. We then used the quotient and the remainder to find the width of the rectangle.

In this article, we will answer some of the most frequently asked questions about synthetic division and its application in finding the width of a rectangle.

Q&A

Q: What is synthetic division?

A: Synthetic division is a method used to divide polynomials by linear factors. It is a shortcut method that eliminates the need for long division.

Q: How do I perform synthetic division?

A: To perform synthetic division, you need to follow these steps:

Write down the coefficients of the polynomial in a row.
Write down the value of the linear factor, which is $c$ in the expression $(x - c)$ .
Bring down the first coefficient.
Multiply the value of the linear factor by the first coefficient and write the result below the second coefficient.
Add the second coefficient and the result from step 4.
Repeat steps 4 and 5 until you reach the last coefficient.
The final result is the quotient and the remainder.

Q: What is the quotient and the remainder in synthetic division?

A: The quotient is the result of dividing the polynomial by the linear factor, and the remainder is the amount left over after the division.

Q: How do I use the quotient and the remainder to find the width of a rectangle?

A: To find the width of a rectangle, you need to use the quotient and the remainder to set up an equation. The equation is:

$A = l \times w$

where $A$ is the area, $l$ is the length, and $w$ is the width.

You can substitute the length into the area formula to get:

$A = (x + 3) \times w$

You can expand the right-hand side of the equation to get:

$A = xw + 3w$

You can equate the coefficients of the two polynomials to get:

$5x^3 + 19x^2 + 6x - 18 = xw + 3w$

You can simplify the equation to get:

$5x^3 + 19x^2 + 6x - 18 = (x + 3)w$

You can divide both sides of the equation by $(x + 3)$ to get:

$w = \frac{5x^3 + 19x^2 + 6x - 18}{x + 3}$

You can use synthetic division to divide the polynomial by $(x + 3)$ to get:

$w = 5x^2 + 4x - 6 + \frac{54}{x + 3}$

Q: What is the final answer?

A: The final answer is $5x^2 + 4x - 6 + \frac{54}{x + 3}$ .

Conclusion

In this article, we answered some of the most frequently asked questions about synthetic division and its application in finding the width of a rectangle. We used synthetic division to divide the polynomial $5x^3 + 19x^2 + 6x - 18$ by the linear factor $(x + 3)$ to get the quotient and the remainder. We then used the quotient and the remainder to find the width of the rectangle.

We hope that this article has been helpful in understanding synthetic division and its application in finding the width of a rectangle. If you have any further questions, please do not hesitate to contact us.