Factoring Polynomials: A Step-by-Step Guide

Hey math enthusiasts! Ever stared at a polynomial and felt like you were looking at a puzzle? Well, you're not alone! Factoring polynomials can seem tricky at first, but once you break it down into manageable steps, it becomes a lot more approachable. In this article, we'll dive deep into the world of factoring, starting with the given polynomial: $2x^5 + 12x^3 - 54x$. We'll explore the process step-by-step, explaining the logic behind each move, and then arrive at the completely factored form. So, buckle up, grab your pencils, and let's unravel this mathematical mystery together! Factoring polynomials is a fundamental skill in algebra, used in solving equations, simplifying expressions, and understanding the behavior of functions. It's like learning to read – once you grasp the basics, you can unlock a whole world of possibilities. Let's get started, shall we?

Step 1: Identify the Greatest Common Factor (GCF)

Alright, guys, our first step is always the same: look for the Greatest Common Factor (GCF). This is the largest factor that divides evenly into all the terms of the polynomial. In our example, $2x^5 + 12x^3 - 54x$, we can see that each term has a '2' and an 'x' in common. So, the GCF is $2x$. We'll factor out this GCF from each term. This is like saying, "What's the biggest thing we can pull out of all these terms?" In this case, it's $2x$. When we do this, we're essentially dividing each term by $2x$ and rewriting the polynomial. This step simplifies the expression and makes it easier to work with. Understanding GCF is critical because it sets the foundation for further factorization. It is like taking the first step to unlock the equation. It's the most basic operation used in almost all mathematical problems. Make sure not to skip this step; it will help you in further steps. Factoring the GCF is often the first and easiest step in factoring any polynomial, so don't miss out on it! Let's get down to the business of the GCF and make our polynomial simpler.

$2x^5 + 12x^3 - 54x = 2x(x^4 + 6x^2 - 27)$

Step 2: Factor the Remaining Polynomial

Now we're left with the expression inside the parentheses: $(x^4 + 6x^2 - 27)$. This looks like a quadratic equation in disguise, doesn't it? We can treat $x^2$ as a single variable. Now, let's focus on factoring the quadratic expression. We need to find two numbers that multiply to -27 (the constant term) and add up to 6 (the coefficient of the $x^2$ term). Let's go through the possible factors: 1 and 27, 3 and 9. Now, if we use 3 and 9, and make the 3 negative and the 9 positive, we get the numbers that fit our condition. The numbers are -3 and 9. Thus, we can rewrite the expression as $(x^2 - 3)(x^2 + 9)$.

This step involves more complex thinking, so you need to be very attentive to understand. You have to understand that the terms are interchangeable. We are using a mathematical trick that helps us to approach the problem more easily. You need to always keep the previous steps in mind. Don't be too hard on yourself if it is confusing. It will be easier with a bit of practice. This is the part that takes practice, so don't be discouraged if it doesn't click immediately. Practice makes perfect, and with each attempt, you'll become more comfortable with the process. Keep in mind that not all polynomials can be factored easily, or at all! But with practice, you will learn to spot the patterns and the techniques that work. Ready? Let's keep going!

So, the polynomial becomes:

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

Step 3: Check for Further Factoring

Okay, before we declare victory, let's take a closer look at our factored expression: $2x(x^2 - 3)(x^2 + 9)$. Can we factor any further? The $2x$ is already in its simplest form. $(x^2 - 3)$ is a difference of squares, but 3 isn't a perfect square. Thus, we can't factor it further using integers. Finally, the term $(x^2 + 9)$ is a sum of squares, and it can't be factored using real numbers. So, this expression is as factored as it can get. So, in this step, you need to identify any further simplifications that may be available to you. You need to see if any of the terms are factorable or not. Now we know how to factor the polynomial completely. To do so, let's sum up everything we learned and see if we can get the solution for the question!

Step 4: Final Answer

Alright, guys! After following all the steps, let's look back at the original question to see if we can find the correct answer! The original expression, $2x^5 + 12x^3 - 54x$, is completely factored into $2x(x^2 - 3)(x^2 + 9)$. This is the fully factored form. So, the correct answer is D. This is the final step; you need to make sure to review all the steps again. It is very important to get the correct answer. The important part is that you understand the process. With that knowledge, you can solve any related question in the future. Now you know how to completely factor this polynomial. Congratulations! You've successfully factored the polynomial! This process may seem a bit long, but as you practice, you'll find yourself able to do these steps more quickly. Remember, the key is to break the problem into smaller steps and take it one step at a time. The more problems you solve, the better you'll get at recognizing patterns and applying the correct techniques.

Therefore, the answer is: D. $2x(x^2 - 3)(x^2 + 9)$