Factoring Common Factors In Polynomial Expressions 8x³ + 12x
Hey guys! Let's dive into a common algebra problem: factoring out the common factor from the expression 8x³ + 12x. This is a fundamental skill in algebra, and mastering it will help you simplify expressions and solve equations more easily. We'll break down the process step-by-step, making sure you understand not just the how, but also the why behind each step. So, grab your pencils, and let's get started!
Understanding Factoring
Before we jump into the problem, let's quickly recap what factoring actually means. In simple terms, factoring is the reverse of expanding. When we expand, we multiply terms together (like using the distributive property). Factoring, on the other hand, involves breaking down an expression into its multiplicative components. Think of it like this: if expanding is like building a house with bricks, factoring is like taking the house apart to see what bricks it's made of.
In the context of our problem, factoring out the common factor means identifying the greatest common factor (GCF) that divides both terms in the expression and then rewriting the expression as a product of the GCF and the remaining terms. This is a powerful technique that simplifies complex expressions and makes them easier to work with. Understanding the basic principle of factoring is crucial before attempting to solve problems like factoring the common factor out of 8x³ + 12x. Remember, the goal is to find the largest expression that divides evenly into all terms, allowing us to rewrite the original expression in a more simplified, factored form. This simplification not only aids in solving equations but also provides a deeper understanding of the structure of algebraic expressions. So, with this foundation in mind, let's move on to the specific steps involved in factoring our given expression.
Identifying the Greatest Common Factor (GCF)
The first key step in factoring 8x³ + 12x is to pinpoint the greatest common factor (GCF) of the terms. The GCF is the largest factor that divides evenly into all terms in the expression. To find it, we need to consider both the coefficients (the numbers) and the variables.
Let's start with the coefficients: 8 and 12. What's the largest number that divides both 8 and 12? If you think about the factors of each number, you'll see that it's 4 (8 = 4 * 2 and 12 = 4 * 3). So, 4 is part of our GCF.
Now, let's look at the variables. We have x³ (which means x * x * x) and x. What's the highest power of x that divides both terms? Well, both terms have at least one x, so x is also part of our GCF.
Combining these two parts, we find that the GCF of 8x³ and 12x is 4x. This means that 4x is the largest expression that can be evenly divided from both terms in our original expression. Identifying the GCF correctly is a cornerstone of factoring, as it sets the stage for simplifying the expression efficiently. A clear understanding of how to break down both numerical coefficients and variable components is crucial for this step. By meticulously examining each term, we ensure that the GCF we extract is indeed the greatest, which leads to the most simplified factored form. This foundational step not only makes the subsequent factoring process smoother but also enhances our ability to recognize and apply similar techniques in more complex algebraic scenarios. So, now that we've successfully identified the GCF, let's move on to the next step: factoring it out.
Factoring Out the GCF
Now that we've identified the greatest common factor (GCF) as 4x, the next step is to factor it out of the expression 8x³ + 12x. This involves dividing each term in the expression by the GCF and writing the result in a factored form.
Here's how we do it:
- Divide the first term, 8x³, by the GCF, 4x: (8x³) / (4x) = 2x². Remember the rules of exponents: when dividing terms with the same base, you subtract the exponents. So, x³ / x = x^(3-1) = x².
- Divide the second term, 12x, by the GCF, 4x: (12x) / (4x) = 3. The x terms cancel out, leaving just the constant 3.
Now, we rewrite the expression in factored form. We take the GCF (4x) and multiply it by the expression formed by the results of our divisions (2x² + 3). This gives us:
4x(2x² + 3)
And that's it! We've successfully factored out the GCF from the original expression. This process of factoring out the GCF is like reverse distribution. We're essentially undoing the distributive property to rewrite the expression in a more simplified and insightful form. Mastering this technique is crucial for tackling more complex algebraic manipulations. It's not just about arriving at the correct answer; it's about understanding how and why this method works. By meticulously dividing each term by the GCF and rewriting the expression, we gain a deeper appreciation for the structure of algebraic expressions and their underlying factors. So, let's take a moment to reflect on what we've accomplished and ensure we're ready to apply this skill in various contexts.
Checking Your Answer
It's always a good idea to check your answer when factoring, especially in mathematics. This ensures that you've factored correctly and haven't made any mistakes. The easiest way to check is to simply distribute the GCF back into the parentheses and see if you get the original expression.
In our case, we factored 8x³ + 12x and got 4x(2x² + 3). To check, we'll distribute 4x across the terms inside the parentheses:
4x * (2x²) = 8x³ 4x * (3) = 12x
Adding these results together, we get:
8x³ + 12x
This is exactly our original expression! This confirms that our factoring is correct. Checking your work by distributing the GCF back into the parentheses is a critical step in the factoring process. It's a quick and effective way to verify that the factored expression is indeed equivalent to the original expression. This practice not only helps in ensuring accuracy but also reinforces the understanding of the relationship between factoring and distribution – two fundamental concepts in algebra. By diligently checking our answers, we build confidence in our skills and develop a more thorough grasp of algebraic manipulations. So, always make it a habit to double-check your factored expressions; it's a simple step that can save you from potential errors and strengthen your algebraic foundation.
Conclusion
Alright, guys! We've successfully factored out the common factor from 8x³ + 12x, and the correct answer is 4x(2x² + 3). We walked through the process step-by-step, from identifying the greatest common factor (GCF) to factoring it out and finally checking our answer. This is a fundamental skill in algebra, and mastering it will help you tackle more complex problems with confidence. Remember, practice makes perfect, so keep working on these types of problems to solidify your understanding. Factoring, at its core, is about simplifying and understanding the structure of algebraic expressions. It's a skill that unlocks doors to more advanced topics in mathematics and beyond. So, embrace the challenge, keep practicing, and watch your algebraic prowess grow. And remember, every problem you solve is a step closer to mastering the art of mathematics. So, keep up the great work, and let's continue exploring the fascinating world of algebra together!
Final Answer: The final answer is (b)
