Factoring $24x^3 - 81$: A Step-by-Step Guide

Hey guys! Let's dive into factoring the expression $24x^3 - 81$. Factoring can seem tricky, but we'll break it down step by step, making it super easy to understand. We'll start by identifying common factors, then apply the difference of cubes pattern if it fits. So, grab your pencils and let's get started!

1. Identifying the Greatest Common Factor (GCF)

When you first encounter an expression like $24x^3 - 81$, your initial step should always be to look for the Greatest Common Factor (GCF). The GCF is the largest number and/or variable that divides evenly into all terms of the expression. In this case, we have two terms: $24x^3$ and $-81$. Let's think about the factors of the coefficients, which are 24 and 81.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 81 are 1, 3, 9, 27, and 81. By comparing these lists, we can see that the greatest common factor between 24 and 81 is 3. Now, let’s consider the variable part. The first term has $x^3$, while the second term is a constant, so there's no common variable factor here.

Therefore, the GCF of $24x^3$ and $-81$ is 3. Factoring out the GCF is like reverse distribution. We divide each term by the GCF and write the GCF outside the parentheses. So, we divide $24x^3$ by 3, which gives us $8x^3$, and we divide -81 by 3, which gives us -27. Now we can rewrite the expression as:

$24x^3 - 81 = 3(8x^3 - 27)$

Factoring out the GCF is super important because it simplifies the expression and makes the subsequent factoring steps much easier. Always remember to look for the GCF first – it’s a game-changer!

2. Recognizing the Difference of Cubes Pattern

Now that we've factored out the GCF, we're left with the expression inside the parentheses: $8x^3 - 27$. This looks like it might fit a special factoring pattern called the difference of cubes. The difference of cubes pattern is a handy tool for factoring expressions in the form $a^3 - b^3$. So, what does this pattern actually look like?

The formula for the difference of cubes is:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

This formula tells us that if we can identify an expression as the difference of two cubes, we can factor it into a binomial $(a - b)$ and a trinomial $(a^2 + ab + b^2)$. To use this pattern, we need to confirm that our expression fits the $a^3 - b^3$ format. Let's take a closer look at $8x^3 - 27$.

Can we express $8x^3$ as something cubed? Absolutely! $8x^3$ is the same as $(2x)^3$ because $2^3 = 8$ and $(x)^3 = x^3$. How about 27? We can also express 27 as a cube, since $27 = 3^3$. So, we can rewrite our expression $8x^3 - 27$ as $(2x)^3 - 3^3$.

Now it’s crystal clear that we have a difference of cubes, where $a = 2x$ and $b = 3$. Recognizing this pattern is half the battle, guys! Once you spot it, applying the formula is the next step, and it’s super straightforward.

3. Applying the Difference of Cubes Formula

Alright, we've identified that our expression $8x^3 - 27$ fits the difference of cubes pattern, where $a = 2x$ and $b = 3$. Now, we're ready to apply the difference of cubes formula:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Let's plug in our values for $a$ and $b$ into the formula. We have $a = 2x$ and $b = 3$, so we substitute these into the formula:

$(2x)^3 - 3^3 = (2x - 3)((2x)^2 + (2x)(3) + 3^2)$

Now, we need to simplify the terms inside the parentheses. First, let's simplify $(2x)^2$. Remember, when you square a product, you square each factor. So, $(2x)^2$ becomes $2^2 * x^2$, which is $4x^2$. Next, let's simplify $(2x)(3)$, which is simply $6x$. Finally, $3^2$ is 9.

So, our expression now looks like this:

$(2x - 3)(4x^2 + 6x + 9)$

We've successfully applied the difference of cubes formula and factored the expression! This step is all about careful substitution and simplification. Take your time, double-check your work, and you'll nail it every time!

4. Combining the GCF and Difference of Cubes Factors

Okay, we've done some serious factoring! We started by factoring out the Greatest Common Factor (GCF) from the original expression, and then we tackled the difference of cubes pattern. Now, it's time to bring it all together and write out the fully factored expression. Remember, the first thing we did was factor out the GCF, which was 3. This gave us:

$24x^3 - 81 = 3(8x^3 - 27)$

Then, we recognized that the expression inside the parentheses, $8x^3 - 27$, could be factored using the difference of cubes pattern. We found that $8x^3 - 27$ factors into $(2x - 3)(4x^2 + 6x + 9)$. So, now we need to substitute this factored form back into our expression.

We replace $8x^3 - 27$ with its factored form, giving us:

$3(8x^3 - 27) = 3(2x - 3)(4x^2 + 6x + 9)$

And that’s it! We've combined the GCF and the difference of cubes factors to completely factor the original expression. This is our final factored form. It’s like putting the pieces of a puzzle together – each step builds on the previous one, leading to the complete solution.

5. Checking for Further Factoring (and Why It Matters)

We've arrived at our factored expression: $3(2x - 3)(4x^2 + 6x + 9)$. But before we declare victory, it's crucial to do one last check: Can we factor any further? This step is often overlooked, but it’s super important to ensure we've factored the expression completely.

We have three factors: 3, $(2x - 3)$, and $(4x^2 + 6x + 9)$. The first factor, 3, is just a constant and can't be factored further. The second factor, $(2x - 3)$, is a linear expression (the highest power of $x$ is 1), and linear expressions can’t be factored unless there's a common factor, which there isn't here. So, what about the third factor, the trinomial $(4x^2 + 6x + 9)$?

This is a quadratic trinomial, and we need to see if it can be factored into two binomials. We could try various factoring techniques, but there's a neat little trick to help us quickly determine if a quadratic trinomial resulting from a difference or sum of cubes can be factored further. Often, these trinomials are prime, meaning they cannot be factored using integer coefficients. This is the case here.

So, after checking each factor, we confirm that none of them can be factored any further. This means we've indeed factored the original expression completely. Checking for further factoring is like the final polish on a masterpiece – it ensures that our solution is not only correct but also in its simplest form. Always take that extra moment to check – it makes all the difference!

Final Answer

So, after all that awesome factoring work, the completely factored form of the expression $24x^3 - 81$ is:

$3(2x - 3)(4x^2 + 6x + 9)$

Great job, guys! Factoring can be a bit of a puzzle, but by breaking it down step by step and remembering key patterns like the difference of cubes, you can conquer any expression that comes your way. Keep practicing, and you'll become a factoring pro in no time! Remember, the key steps were to identify and factor out the GCF, recognize the difference of cubes pattern, apply the formula, combine the factors, and double-check for any further factoring. You nailed it!