Find Rectangular Prism Base Area Using Synthetic Division

Hey guys, welcome back to another math adventure! Today, we're diving deep into the awesome world of algebraic expressions and geometric shapes. We've got a cool problem that combines the power of synthetic division with the concept of a rectangular prism. Our mission, should we choose to accept it, is to find the expression for the area of the base of a rectangular prism. We're given its height, which is $x+4$, and its volume, a snazzy cubic expression: $x^3+2 x^2-17 x-36$. Stick around, because by the end of this, you'll be a synthetic division whiz and understand how it helps us solve real-world (well, math-world) problems!

Unpacking the Rectangular Prism and Volume Formula

Alright, let's get down to business. First off, what exactly is a rectangular prism? Think of a box, like a cereal box or a shoebox. It's a 3D shape with six rectangular faces. The volume of any prism, and a rectangular prism is no exception, is found by multiplying the area of its base by its height. So, the trusty formula is: Volume = Base Area × Height. In this problem, we're given the volume $V = x^3+2 x^2-17 x-36$ and the height $h = x+4$. We need to find the Base Area. Using our formula, we can rearrange it to solve for the Base Area: Base Area = Volume / Height.

This means we need to perform a division. We're dividing the polynomial representing the volume by the polynomial representing the height. Now, we could use long division, but where's the fun in that? This is where our star player, synthetic division, comes in. Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form $(x-c)$. Since our height is $x+4$, we can rewrite it as $x - (-4)$. This means our $c$ value for synthetic division is $-4$. This method is super efficient and can save us a ton of time, especially when dealing with higher-degree polynomials. So, we're going to use synthetic division to divide $x^3+2 x^2-17 x-36$ by $x+4$. The result of this division will be the expression for the area of the base.

Mastering Synthetic Division: Step-by-Step

Now, let's get our hands dirty with synthetic division. It's a systematic process that helps us divide polynomials quickly. Here's how we tackle our problem: $(x^3+2 x^2-17 x-36) \div (x+4)$.

Step 1: Set up the synthetic division.

We'll use the root of the divisor $(x+4)$, which is $-4$, as our outside number. Then, we write down the coefficients of the dividend polynomial $x^3+2 x^2-17 x-36$ in order from the highest power of $x$ to the lowest. Make sure to include a zero for any missing terms (though we don't have any missing terms here).

The coefficients are: $1$ (for $x^3$), $2$ (for $x^2$), $-17$ (for $x$), and $-36$ (for the constant term).

So, our setup looks like this:

-4 | 1   2   -17   -36
   |__________________

Step 2: Bring down the first coefficient.

Bring the leading coefficient (which is $1$) straight down below the line.

-4 | 1   2   -17   -36
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     1

Step 3: Multiply and add.

Now, multiply the number you just brought down ($1$) by the outside number ($-4$). This gives us $-4$. Write this result under the next coefficient ($2$). Then, add the numbers in that column ($2 + (-4) = -2$).

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

Step 4: Repeat the multiply and add process.

Take the result from the previous addition ($-2$) and multiply it by the outside number ($-4$). This gives us $8$. Write $8$ under the next coefficient ($-17$). Add the numbers in this column ($-17 + 8 = -9$).

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

Step 5: One more time!

Take the result from the last addition ($-9$) and multiply it by the outside number ($-4$). This gives us $36$. Write $36$ under the last coefficient ($-36$). Add the numbers in this column ($-36 + 36 = 0$).

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

Step 6: Interpret the results.

The numbers below the line, except for the last one, are the coefficients of the quotient polynomial. The last number is the remainder. In our case, the coefficients are $1$, $-2$, and $-9$, and the remainder is $0$. Since the original polynomial was degree 3, the quotient will be degree 2. So, the quotient is $1x^2 - 2x - 9$. The remainder is $0$, which is exactly what we want when dividing the volume by the height to find the base area. A remainder of $0$ means that $(x+4)$ is a factor of the volume polynomial, and the quotient is the other factor, which represents the base area.

So, the expression for the area of the base is $x^2 - 2x - 9$.

Connecting Synthetic Division to the Base Area

Awesome job, guys! We've successfully used synthetic division to divide the volume of the rectangular prism by its height. Remember, the volume of a rectangular prism is given by $V = ext{Base Area} imes ext{Height}$. When we divide the volume by the height, we are essentially isolating the expression for the Base Area. Our synthetic division calculation showed us that:

$$\frac{x^3+2 x^2-17 x-36}{x+4} = x^2 - 2x - 9$$

This means that the Base Area of our rectangular prism is given by the expression $x^2 - 2x - 9$. This is a quadratic expression, which makes sense because the base of a rectangular prism is a rectangle, and the area of a rectangle is found by multiplying its length and width (which are likely linear expressions in $x$ if their product is a quadratic).

Let's quickly check our answer. If the Base Area is $x^2 - 2x - 9$ and the Height is $x+4$, then the Volume should be their product:

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

We can expand this using the distributive property (or FOIL, but extended for a binomial and trinomial):

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

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

Now, combine like terms:

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

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

Voila! This matches the original volume expression given in the problem. This confirms that our synthetic division was accurate and that $x^2 - 2x - 9$ is indeed the correct expression for the area of the base.

The Options and Our Solution

Now, let's look at the options provided:

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

Comparing our result, $x^2 - 2x - 9$, with these options, we can clearly see that it matches option C. So, the expression for the area of the base of the rectangular prism is $x^2 - 2x - 9$.

Why Synthetic Division is Your Friend

So, why bother with synthetic division? Well, as you saw, it's a much faster way to divide polynomials by linear binomials compared to traditional long division. Think about it: fewer steps, less writing, and a reduced chance of making silly arithmetic errors. When you're dealing with potentially complex problems like finding dimensions or areas of geometric shapes using algebraic expressions, efficiency matters. Synthetic division is a powerful tool in your algebraic toolkit. It's particularly useful when you need to find factors of polynomials, check for roots (since a remainder of 0 means the value you tested is a root), or, as in this case, solve problems involving volumes and dimensions.

Remember, the mathematics behind these problems often involves understanding fundamental concepts like the volume of a prism and then applying efficient techniques like synthetic division to solve them. This problem beautifully illustrates how these concepts work together. The expression for the area of the base is a direct result of performing the correct polynomial division. Keep practicing synthetic division, and you'll find it becomes second nature. It's all about breaking down complex problems into manageable steps, and synthetic division is a fantastic shortcut for one of those crucial steps!

Keep exploring, keep questioning, and keep mastering those math skills, guys! See you in the next one!