Factored Form Of 2x^3 + 4x^2 - X Step By Step Solution

Hey there, math enthusiasts! Ever stumbled upon a polynomial expression and felt a little lost trying to break it down? Don't worry, we've all been there. Today, we're going to tackle a classic factoring problem: finding the factored form of the expression 2x³ + 4x² - x. We'll walk through the steps together, making sure you understand not just the how, but also the why behind each move. So, let's dive in and make factoring feel like a breeze!

Understanding Factoring: The Key to Simplifying Expressions

Before we jump into the specific problem, let's take a moment to refresh our understanding of factoring. In essence, factoring is the reverse of expanding. Think of it like this: when you expand, you multiply terms together to get a larger expression. When you factor, you break down a larger expression into smaller, multiplied terms. Factoring is an essential skill in algebra and beyond. It allows us to simplify complex expressions, solve equations, and gain deeper insights into mathematical relationships. Imagine trying to solve a quadratic equation without factoring – it would be a much tougher task! Factoring polynomials is like finding the building blocks that make up a mathematical structure. By identifying these fundamental components, we can better understand the overall behavior and properties of the expression. For example, factoring helps us determine the roots or zeros of a polynomial, which are the values of x that make the expression equal to zero. These roots provide crucial information about the graph of the polynomial function, such as where it intersects the x-axis. Moreover, factoring plays a significant role in calculus, where it is used to simplify expressions before differentiation or integration. Factoring also comes in handy when dealing with rational expressions, allowing us to simplify fractions by canceling out common factors in the numerator and denominator. So, you see, factoring isn't just a standalone skill; it's a versatile tool that unlocks doors to various mathematical concepts and applications. In the world of polynomials, the greatest common factor (GCF) is a crucial concept. It's the largest factor that divides all terms in a polynomial. Think of it as the common thread that ties the terms together. Identifying and extracting the GCF is often the first and most important step in factoring, as it simplifies the expression and reveals its underlying structure. For instance, in our problem, the GCF is x, which we'll factor out in the next step. Remember, factoring is a skill that improves with practice. The more you work with different types of expressions, the better you'll become at recognizing patterns and applying appropriate techniques. So, don't be discouraged if you find it challenging at first. Keep practicing, and you'll soon master the art of factoring!

Step-by-Step Solution: Factoring 2x³ + 4x² - x

Now, let's tackle our specific problem. We're looking for the factored form of the polynomial 2x³ + 4x² - x. The first thing we always want to do when factoring is to look for a greatest common factor (GCF). What's the largest factor that divides all the terms in our expression? Looking at 2x³, 4x², and -x, we can see that each term has at least one x. So, x is our GCF. Factoring out the x means we divide each term by x and write the result in parentheses: x(2x² + 4x - 1). Great! We've taken the first step and simplified our expression. Now we need to see if the expression inside the parentheses, 2x² + 4x - 1, can be factored further. This is a quadratic expression, so we're looking for two binomials that multiply to give us this quadratic. To factor a quadratic in the form ax² + bx + c, we typically look for two numbers that multiply to ac and add up to b. In our case, a = 2, b = 4, and c = -1. So, we need two numbers that multiply to (2)(-1) = -2 and add up to 4. Hmmm... let's think about the factors of -2. We have -1 and 2, or 1 and -2. None of these pairs add up to 4. This tells us that the quadratic 2x² + 4x - 1 cannot be factored further using simple integer factors. It's what we call a prime quadratic. So, we've reached the end of our factoring journey for this expression. We factored out the GCF, and the remaining quadratic couldn't be factored further. That means our final factored form is: x(2x² + 4x - 1). And that's it! We've successfully factored the polynomial. Remember, factoring is like a puzzle – you need to find the right pieces and put them together in the correct way. Always start by looking for the GCF, and then see if you can factor the remaining expression further. With practice, you'll become a factoring pro!

Identifying the Correct Answer: Matching Our Result

Alright, we've factored the expression 2x³ + 4x² - x and arrived at the factored form: x(2x² + 4x - 1). Now, let's take a look at the answer choices provided and see which one matches our result. We have the following options:

A. 2x(x² + 2x + 1) B. x(2x² + 4x + 1) C. 2x(x² + 2x - 1) D. x(2x² + 4x - 1)

Comparing our factored form x(2x² + 4x - 1) with the answer choices, we can clearly see that option D is an exact match. Option A has a GCF of 2x, which is incorrect. Option B has a different constant term inside the parentheses (+1 instead of -1). Option C also has an incorrect GCF. Therefore, the correct answer is D. x(2x² + 4x - 1). It's always a good practice to double-check your work, especially when you have answer choices to compare against. This helps you catch any potential errors and ensures that you've arrived at the correct solution. In this case, we've confidently identified the correct answer by carefully factoring the expression and matching it with the provided options. Remember, the key to solving factoring problems is to follow a systematic approach: look for the GCF first, then try to factor the remaining expression further, and finally, double-check your answer. With these steps in mind, you'll be well-equipped to tackle any factoring challenge that comes your way!

Common Factoring Mistakes and How to Avoid Them

Factoring can be tricky, and it's easy to make mistakes if you're not careful. Let's talk about some common pitfalls and how to avoid them. One of the most frequent errors is forgetting to factor out the greatest common factor (GCF) first. This can lead to more complex factoring later on, or even an incorrect answer. Always, always, always look for the GCF before you do anything else! In our problem, if we hadn't factored out the x initially, we would have been trying to factor the more complicated expression 2x³ + 4x² - x directly, which is much harder. Another common mistake is incorrectly distributing the GCF back in to check your answer. Remember, factoring is the reverse of expanding. So, after you've factored an expression, you can always multiply the factors back together to see if you get the original expression. If you don't, you've made a mistake somewhere. For example, if we factored 2x³ + 4x² - x as x(2x² + 4x - 1), we can multiply x by each term inside the parentheses to verify that we get back to the original expression. A third mistake is incorrectly factoring quadratic expressions. This often happens when people try to guess the factors without using a systematic approach. Remember the