Factoring $17x^4y^2 + 51x^5y: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of factoring algebraic expressions. Specifically, we'll tackle the expression $17x^4y^2 + 51x^5y$. Factoring might seem a bit intimidating at first, but trust me, with the right approach, it becomes a piece of cake. This step-by-step guide will walk you through the process, making sure you grasp every detail. Let's get started, shall we?

Understanding the Basics of Factoring: A Quick Refresher

Before we jump into the main problem, let's quickly recap what factoring is all about. Basically, factoring involves breaking down an expression into a product of simpler expressions (usually polynomials). It's like finding the ingredients (factors) that, when multiplied together, give you the original recipe (the expression). In essence, factoring is the reverse of distribution. When we distribute, we're multiplying something out. When we factor, we're undoing that multiplication. The main goal is to simplify and rewrite the expression in a way that reveals its structure more clearly. It is also often used to solve equations.

There are several factoring techniques, like greatest common factor (GCF), difference of squares, and factoring trinomials. For our specific problem, we'll primarily use the greatest common factor method. This method involves finding the largest expression that divides evenly into all terms in the expression. Think of it as finding the biggest common element among all the terms. This technique is often the first step in simplifying more complex algebraic expressions. Remember, the ultimate aim is to make the expression simpler, and factoring is a powerful tool to achieve this. By the way, always check if your final factored expression can be factored further. If it can, then you are not done yet! Factoring completely means to break it down into the most simplified factors possible. So, keep an eye out for further simplification opportunities.

Now, let's move on to our main objective: factoring the given expression.

Step-by-Step: Factoring $17x^4y^2 + 51x^5y$

Alright, let's break down the expression $17x^4y^2 + 51x^5y$. We'll use the greatest common factor (GCF) method, which means we need to find the largest factor that divides evenly into both terms.

Step 1: Identify the GCF of the Coefficients

First, let's look at the coefficients: 17 and 51. The greatest common factor of 17 and 51 is 17. Because 17 divides evenly into both 17 and 51.

So, we can factor out 17.

Step 2: Identify the GCF of the Variable Terms

Now, let's look at the variable terms. We have $x^4y^2$ and $x^5y$. To find the GCF of the variable terms, we need to look at the powers of each variable.

For x: We have $x^4$ and $x^5$. The lowest power is 4, so the GCF for x is $x^4$.

For y: We have $y^2$ and y. The lowest power is 1, so the GCF for y is y.

Therefore, the GCF of the variable terms is $x^4y$.

Step 3: Factor Out the GCF

Now that we've found the GCF of both the coefficients and the variables, we can factor it out of the expression. The GCF of the entire expression is $17x^4y$.

So, we write:

$17x^4y^2 + 51x^5y = 17x^4y ( ... )$.

To find what goes inside the parentheses, we divide each term of the original expression by the GCF, $17x^4y$.

• For the first term, $17x^4y^2$ divided by $17x^4y$ equals y.
• For the second term, $51x^5y$ divided by $17x^4y$ equals $3x$.

So, we get:

$17x^4y^2 + 51x^5y = 17x^4y(y + 3x)$.

Step 4: Verify Your Answer

Always, always, always verify your answer by distributing the factored expression. Distributing $17x^4y(y + 3x)$ should get us back to the original expression, $17x^4y^2 + 51x^5y$.

• $17x^4y * y = 17x^4y^2$
• $17x^4y * 3x = 51x^5y$

Adding these results together, we get $17x^4y^2 + 51x^5y$, which is our original expression. This confirms that our factoring is correct!

The Final Answer: Factoring Completed!

So, guys, the completely factored form of $17x^4y^2 + 51x^5y$ is $17x^4y(y + 3x)$. We successfully factored the expression by identifying the GCF, factoring it out, and simplifying. Remember the steps: find the GCF of the coefficients, find the GCF of the variables, factor out the GCF, and always verify your work. Factoring helps us understand algebraic expressions more deeply, and it's a super useful skill in algebra. Keep practicing, and you'll become a factoring pro in no time!

Tips and Tricks for Factoring

Want to level up your factoring skills? Here are some extra tips and tricks:

• Always Look for a GCF First: This is the golden rule! Factoring out the GCF is often the easiest first step and simplifies the rest of the process. Even if you see other factoring patterns, start by looking for the GCF.
• Check for Special Patterns: Keep an eye out for special patterns like the difference of squares ($a^2 - b^2 = (a+b)(a-b)$) or perfect square trinomials ($a^2 + 2ab + b^2 = (a+b)^2$). Recognizing these patterns can speed up the factoring process.
• Practice, Practice, Practice: The more you factor, the better you'll become. Work through various examples to get comfortable with different types of expressions.
• Don't Be Afraid to Use Trial and Error: Sometimes, especially with trinomials, you might need to try a few different combinations before you find the right factors. Don't worry, it's part of the process!
• Double-Check Your Work: Always distribute the factored expression to ensure it matches the original expression. This is a crucial step to avoid mistakes.

Conclusion: Mastering Factoring

Factoring can be a challenge, but by breaking it down step by step and understanding the concepts, anyone can master it. This article guided you through factoring a specific expression, highlighting how to identify the greatest common factor and apply it. Remember, practice is key. Keep working through examples, and you'll become confident in your ability to factor any algebraic expression that comes your way. Keep up the great work, and happy factoring, everyone!