Synthetic Division: Find Rectangular Prism Base Area

Hey math enthusiasts! Today, we're diving into a super cool problem that combines algebra with geometry. We're going to use a powerful tool called synthetic division to figure out the expression for the area of the base of a rectangular prism. Imagine you have a box, and you know its volume and its height. Your mission, should you choose to accept it, is to find the area of the bottom surface of that box. Sounds intriguing, right? Let's break it down.

Understanding the Problem: Volume, Height, and Base Area

In the world of rectangular prisms, the relationship between volume, base area, and height is pretty straightforward. The volume of any prism is essentially the area of its base multiplied by its height. Think of it like stacking up layers of the base area. So, if we have the volume ($V$) and the height ($h$), we can find the area of the base ($A$) by rearranging the formula: $A = V / h$. Our problem gives us the volume as a cubic polynomial: $x^3 + 2x^2 - 17x - 36$, and the height as a linear expression: $x + 4$. Our goal is to find the expression for the base area, which we expect to be a quadratic polynomial since Volume (cubic) divided by Height (linear) should result in a quadratic expression.

Why Synthetic Division is Your Go-To Tool

Now, you might be thinking, "Can't I just do regular polynomial long division?" And the answer is yes, you absolutely can! However, synthetic division offers a more streamlined and often quicker method, especially when you're dividing by a linear factor of the form $(x - c)$. It's a clever shortcut that helps us avoid much of the tedious writing involved in long division. It works by using the root of the divisor to perform the division process more compactly. In our case, the divisor is $(x + 4)$, so the root we'll be using is $-4$. This method is incredibly useful in algebra for finding roots of polynomials and for simplifying rational expressions, like the one we're dealing with to find the base area. It's like having a special tool in your mathematical toolbox that makes certain jobs significantly easier.

Step-by-Step: Performing Synthetic Division

Let's get down to business and perform the synthetic division. We need to divide the volume polynomial ($x^3 + 2x^2 - 17x - 36$) by the height expression $(x + 4)$. Remember, for synthetic division, we use the root of the divisor. Since our divisor is $(x + 4)$, the root is $-4$.

First, set up the synthetic division. Write the root ($-4$) to the left. Then, write down the coefficients of the volume polynomial in order of descending powers of $x$: $1$ (for $x^3$), $2$ (for $x^2$), $-17$ (for $x$), and $-36$ (the constant term).

-4 | 1   2   -17   -36
   |_________________

Now, bring down the first coefficient (which is $1$) below the line.

-4 | 1   2   -17   -36
   |_________________
     1

Next, multiply the number you just brought down ($1$) by the root ($-4$) and write the result ($-4$) under the next coefficient ($2$).

-4 | 1   2   -17   -36
   |    -4
   |_________________
     1

Add the numbers in the second column ($2$ and $-4$) and write the sum ($-2$) below the line.

-4 | 1   2   -17   -36
   |    -4
   |_________________
     1  -2

Repeat the process: multiply the new number below the line ($-2$) by the root ($-4$) and write the result ($8$) under the next coefficient ($-17$).

-4 | 1   2   -17   -36
   |    -4     8
   |_________________
     1  -2

Add the numbers in the third column ($-17$ and $8$) and write the sum ($-9$) below the line.

-4 | 1   2   -17   -36
   |    -4     8
   |_________________
     1  -2    -9

One more time: multiply the latest number below the line ($-9$) by the root ($-4$) and write the result ($36$) under the last coefficient ($-36$).

-4 | 1   2   -17   -36
   |    -4     8    36
   |_________________
     1  -2    -9

Finally, add the numbers in the last column ($-36$ and $36$) and write the sum ($0$) below the line. This last number is the remainder.

-4 | 1   2   -17   -36
   |    -4     8    36
   |_________________
     1  -2    -9     0

Interpreting the Results: Finding the Base Area

The numbers below the line, excluding the remainder, are the coefficients of the quotient, which represents the area of the base. The remainder is $0$, which is excellent news! It means that $(x + 4)$ is indeed a factor of the volume polynomial, and our division is exact. The coefficients we obtained are $1$, $-2$, and $-9$. Since we started with a cubic polynomial (degree 3) and divided by a linear polynomial (degree 1), the quotient will be a quadratic polynomial (degree 2).

So, the coefficients $1$, $-2$, and $-9$ correspond to the terms $1x^2$, $-2x$, and $-9$. Therefore, the expression for the area of the base is $x^2 - 2x - 9$. This is our answer!

Checking Our Work and the Options

It's always a good idea to double-check our work, especially in mathematics. We found that the area of the base is $x^2 - 2x - 9$. To verify, we can multiply this expression by the height $(x + 4)$ to see if we get the original volume polynomial.

$(x^2 - 2x - 9)(x + 4)$

Let's distribute:

$x(x^2 - 2x - 9) + 4(x^2 - 2x - 9)$

$(x^3 - 2x^2 - 9x) + (4x^2 - 8x - 36)$

Now, combine like terms:

$x^3 + (-2x^2 + 4x^2) + (-9x - 8x) - 36$

$x^3 + 2x^2 - 17x - 36$

This matches the original volume polynomial! So, our calculation using synthetic division is correct.

Now, let's look at the given options:

A. $2 x^2-9 x$ B. $x^2+6 x-24$ C. $x^2-2 x-9$ D. $-4 x^2+8 x+36$

Our calculated area of the base, $x^2 - 2x - 9$, perfectly matches Option C.

Conclusion: The Power of Synthetic Division

We've successfully used synthetic division to tackle a problem involving geometric volumes and algebraic expressions. This technique proved to be an efficient way to divide polynomials, allowing us to find the unknown base area of the rectangular prism. Remember, the key steps are to correctly set up the synthetic division with the root of the divisor and the coefficients of the dividend, and then to carefully interpret the resulting coefficients as the terms of the quotient. This method is not only useful for problems like this but also forms a fundamental building block for more advanced algebraic concepts.

If you're looking to deepen your understanding of polynomial division and synthetic division, I highly recommend checking out resources like Khan Academy, which offers a wealth of free tutorials and practice problems on various mathematics topics. For more advanced mathematical concepts and theorems, the Wolfram MathWorld website is an invaluable resource for detailed explanations and definitions.