Factored Form: Unraveling $8x^3 - 8x^2 - 30x$

Hey math enthusiasts! Today, we're diving into the world of factoring – specifically, finding the factored form of the expression: $8x^3 - 8x^2 - 30x$. Factoring might seem a little intimidating at first, but trust me, with a little practice and the right approach, it becomes a super useful tool in your mathematical arsenal. So, let's break it down step by step and make this process crystal clear. Our goal is to rewrite this expression as a product of simpler expressions (factors). This skill is crucial for solving equations, simplifying complex expressions, and understanding the behavior of functions. Let's get started, shall we?

Step 1: Identify the Greatest Common Factor (GCF)

Alright, guys, the very first thing we always want to do when factoring is to look for the Greatest Common Factor (GCF). The GCF is the largest factor that divides evenly into all terms of the expression. In our expression, $8x^3 - 8x^2 - 30x$, we need to examine the coefficients (the numbers in front of the variables) and the variables themselves.

Looking at the coefficients: 8, -8, and -30. The largest number that divides evenly into all three is 2. So, 2 is part of our GCF. Now, let's look at the variables. We have $x^3$, $x^2$, and $x$. Each term has at least one x in it. Therefore, x is also part of our GCF. Combining these, our GCF is 2x. We'll factor out 2x from each term. This means we're essentially dividing each term by 2x.

Let's do it! When we divide $8x^3$ by $2x$, we get $4x^2$. Dividing $-8x^2$ by $2x$, we get $-4x$. And dividing $-30x$ by $2x$, we get $-15$. So, we can rewrite our expression as $2x(4x^2 - 4x - 15)$. We've successfully factored out the GCF! Awesome, right? This is a great starting point, and it simplifies our expression significantly. Remember, finding the GCF is often the first and most important step in factoring.

Now, let's check our work. To make sure we've factored correctly, you can always distribute the GCF back into the parentheses. If you get back your original expression, you've done it right. In our case, $2x * 4x^2 = 8x^3$, $2x * -4x = -8x^2$, and $2x * -15 = -30x$. We're good to go!

Step 2: Factor the Quadratic Expression

Okay, team, now we're looking at the quadratic expression inside the parentheses: $4x^2 - 4x - 15$. Our goal is to factor this quadratic into two binomials, if possible. Factoring quadratics can take a few different approaches, and the one you choose might depend on your comfort level and the specific problem. Here, we'll use a method that works by looking for two numbers that multiply to give us the product of the first and last terms (in this case, $4 * -15 = -60$) and add up to the middle term's coefficient (which is -4).

Let's brainstorm some factor pairs of -60: (1, -60), (-1, 60), (2, -30), (-2, 30), (3, -20), (-3, 20), (4, -15), (-4, 15), (5, -12), (-5, 12), (6, -10), and (-6, 10). Now we check which of these pairs add up to -4. After examining the pairs, we find that 6 and -10 fit the bill because 6 * -10 = -60 and 6 + (-10) = -4. Knowing this, we can now rewrite the middle term (-4x) of our quadratic using these two numbers. We rewrite the quadratic expression $4x^2 - 4x - 15$ as $4x^2 + 6x - 10x - 15$. Notice that we have not changed the overall value of the expression, just how it looks.

Next, we'll factor by grouping. We group the first two terms and the last two terms and factor out the GCF from each group. From the first group ($4x^2 + 6x$), the GCF is 2x. Factoring this out, we get $2x(2x + 3)$. From the second group ($-10x - 15$), the GCF is -5. Factoring this out, we get $-5(2x + 3)$. Now our expression is $2x(2x + 3) - 5(2x + 3)$.

Notice that both terms now have a common factor of (2x + 3). We can factor this out. Therefore, we get $(2x + 3)(2x - 5)$.

Step 3: Putting it All Together

Alright, folks, we're in the home stretch! We've done the heavy lifting, and now it's time to assemble all the pieces. Remember that we started with the expression $8x^3 - 8x^2 - 30x$ and, in the first step, factored out the GCF of 2x, which gave us $2x(4x^2 - 4x - 15)$. Then, in the second step, we factored the quadratic expression $4x^2 - 4x - 15$ into $(2x + 3)(2x - 5)$. Now, we'll put it all back together to obtain our final factored form. We take our initial GCF factor of 2x and multiply it by the two binomial factors we obtained from factoring the quadratic expression, $(2x + 3)(2x - 5)$.

Therefore, the complete factored form of the original expression, $8x^3 - 8x^2 - 30x$, is $2x(2x + 3)(2x - 5)$. And that, my friends, is how it's done! We've successfully factored the expression. Isn't that an awesome feeling?

Step 4: Verification

To make sure we've got the correct factored form, we can always check our answer by multiplying everything back out. Let's start by multiplying the two binomials: $(2x + 3)(2x - 5)$. Using the FOIL method (First, Outer, Inner, Last), we get: $(2x * 2x) + (2x * -5) + (3 * 2x) + (3 * -5)$. Which simplifies to $4x^2 - 10x + 6x - 15$. Combining like terms, we get $4x^2 - 4x - 15$. Now, multiply this result by the GCF, 2x: $2x(4x^2 - 4x - 15) = 8x^3 - 8x^2 - 30x$.

As you can see, when we expand our factored form, we get back the original expression, confirming that we've factored correctly. Woohoo! This verification step is a crucial habit to develop, as it allows you to catch any mistakes you might have made along the way. Factoring can be tricky, but this process gives you confidence in your answer.

Tips and Tricks for Factoring

Here are some tips to help you in your factoring journey:

• Always look for the GCF first. This is the golden rule! It simplifies the expression and makes the rest of the factoring process easier.
• Practice, practice, practice! The more you factor, the better you'll become at recognizing patterns and choosing the right factoring methods. Work through different examples to gain familiarity.
• Know your perfect squares and cubes. Recognizing these patterns can speed up the factoring process. For example, knowing that 9 is a perfect square (3^2) can help you identify a difference of squares.
• Don't be afraid to try different methods. There are various ways to factor quadratic expressions (e.g., the quadratic formula), and what works best depends on the specific problem. Experiment to find what you like.
• Check your answer. As we did above, always verify your factored form by multiplying it back out to ensure it matches the original expression.

Final Thoughts

Factoring is a fundamental skill in algebra and is essential for success in higher-level math courses. By mastering this concept, you'll be well-equipped to tackle more complex problems. Remember to always start by looking for the GCF, then factor the remaining expression using appropriate methods. Practice regularly, and don't be discouraged if it takes some time to grasp. Keep at it, and you'll become a factoring pro in no time! Keep practicing, and you'll find that factoring becomes easier and more intuitive with each problem you solve. Happy factoring, everyone!