Factoring $21 W^6 Y^4 Z^5-15 W^2 Y^7 Z^8$ A Step-by-Step Guide

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Hey guys! Today, we're diving into the world of factoring expressions, and we're going to tackle a specific problem: factoring $21 w^6 y^4 z^5-15 w^2 y^7 z^8$. Factoring might sound intimidating, but trust me, it's like solving a puzzle, and once you get the hang of it, it can be pretty fun. So, let's break it down step by step and make sure we understand every piece of the puzzle.

Understanding Factoring

Before we jump into the problem, let's quickly recap what factoring is all about. In simple terms, factoring is like reverse multiplication. Think of it this way: when you multiply two numbers (or expressions) together, you get a product. Factoring is the process of finding those original numbers (or expressions) that multiply together to give you a particular product. For example, if we have the number 12, we can factor it into 3 and 4 because 3 * 4 = 12. Similarly, we can factor algebraic expressions by identifying common factors.

The main goal in factoring is to simplify expressions and break them down into their most basic components. This is super useful in algebra because it helps us solve equations, simplify fractions, and understand the structure of expressions better. When we talk about factoring algebraic expressions, we're looking for terms that divide evenly into all parts of the expression. These terms can be numbers (like 2, 3, 5) or variables (like x, y, z) or even combinations of both (like 2x, 3y, 5z).

So, why is factoring so important? Well, factoring is like having a secret weapon in your math arsenal. It allows you to simplify complex problems and make them much easier to handle. For instance, when you're solving quadratic equations, factoring can help you find the roots quickly. It’s also essential when you're dealing with rational expressions (fractions with polynomials) because it helps you simplify them and identify any common factors that can be canceled out. Plus, factoring is a fundamental concept that you'll use in higher-level math courses like calculus, so mastering it now will definitely pay off in the long run.

Step 1: Identify the Greatest Common Factor (GCF)

Okay, let's get our hands dirty with the expression $21 w^6 y^4 z^5-15 w^2 y^7 z^8$. The first thing we need to do is identify the Greatest Common Factor (GCF). The GCF is the largest factor that divides evenly into all terms of the expression. It's like finding the biggest piece that fits into all the parts of our puzzle.

To find the GCF, we'll look at the coefficients (the numbers in front of the variables) and the variables separately. Let's start with the coefficients: we have 21 and -15. We need to find the largest number that divides both 21 and 15. The factors of 21 are 1, 3, 7, and 21. The factors of 15 are 1, 3, 5, and 15. The largest number they have in common is 3. So, the numerical part of our GCF is 3.

Now, let's move on to the variables. We have $w^6$ and $w^2$, $y^4$ and $y^7$, and $z^5$ and $z^8$. For each variable, we'll take the lowest exponent that appears in the terms. This is because the GCF can only contain powers of variables that are present in all terms, and the exponent has to be the smallest one to ensure it divides evenly into all terms.

  • For w, we have $w^6$ and $w^2$. The lowest exponent is 2, so we'll include $w^2$ in our GCF.
  • For y, we have $y^4$ and $y^7$. The lowest exponent is 4, so we'll include $y^4$ in our GCF.
  • For z, we have $z^5$ and $z^8$. The lowest exponent is 5, so we'll include $z^5$ in our GCF.

Putting it all together, our GCF is $3 w^2 y^4 z^5$. This is the biggest piece that we can factor out of both terms in our expression. Remember, the GCF is the key to simplifying the expression and making it easier to work with. Finding the GCF might seem like a small step, but it's a crucial one because it sets the stage for the rest of the factoring process. If you mess up the GCF, the rest of your factoring will be off, so take your time and make sure you get it right!

Step 2: Factor Out the GCF

Alright, now that we've identified our Greatest Common Factor (GCF) as $3 w^2 y^4 z^5$, it's time to actually factor it out of our expression $21 w^6 y^4 z^5-15 w^2 y^7 z^8$. Factoring out the GCF is like pulling out the biggest common piece from each term, leaving behind what's left after the division.

To do this, we'll divide each term in the expression by the GCF. Let's start with the first term, $21 w^6 y^4 z^5$. We're going to divide this by $3 w^2 y^4 z^5$:

21w6y4z53w2y4z5\frac{21 w^6 y^4 z^5}{3 w^2 y^4 z^5}

First, divide the coefficients: 21 ÷ 3 = 7.

Next, divide the variables. Remember the rule for dividing exponents: you subtract the exponents. So:

  • w6÷w2=w6−2=w4w^6 ÷ w^2 = w^{6-2} = w^4

  • y4÷y4=y4−4=y0=1y^4 ÷ y^4 = y^{4-4} = y^0 = 1

  • z5÷z5=z5−5=z0=1z^5 ÷ z^5 = z^{5-5} = z^0 = 1

So, when we divide $21 w^6 y^4 z^5$ by $3 w^2 y^4 z^5$, we get $7 w^4$. This is what's left of the first term after we've factored out the GCF.

Now, let's move on to the second term, $-15 w^2 y^7 z^8$. We'll divide this by $3 w^2 y^4 z^5$:

−15w2y7z83w2y4z5\frac{-15 w^2 y^7 z^8}{3 w^2 y^4 z^5}

Again, start with the coefficients: -15 ÷ 3 = -5.

Now, divide the variables:

  • w2÷w2=w2−2=w0=1w^2 ÷ w^2 = w^{2-2} = w^0 = 1

  • y7÷y4=y7−4=y3y^7 ÷ y^4 = y^{7-4} = y^3

  • z8÷z5=z8−5=z3z^8 ÷ z^5 = z^{8-5} = z^3

So, when we divide $-15 w^2 y^7 z^8$ by $3 w^2 y^4 z^5$, we get $-5 y^3 z^3$. This is what's left of the second term after factoring out the GCF.

Now that we've divided both terms by the GCF, we can rewrite the original expression as the GCF multiplied by the result of our divisions:

21w6y4z5−15w2y7z8=3w2y4z5(7w4−5y3z3)21 w^6 y^4 z^5-15 w^2 y^7 z^8 = 3 w^2 y^4 z^5 (7 w^4 - 5 y^3 z^3)

So, we've successfully factored out the GCF! The expression $3 w^2 y^4 z^5 (7 w^4 - 5 y^3 z^3)$ is the factored form of our original expression. We've broken it down into two main parts: the GCF and the remaining expression inside the parentheses. Factoring out the GCF is a powerful technique because it simplifies the expression and makes it easier to work with. It's like taking a big, messy problem and organizing it into smaller, more manageable pieces.

Step 3: Check for Further Factoring

We've factored out the Greatest Common Factor (GCF), and we've got $3 w^2 y^4 z^5 (7 w^4 - 5 y^3 z^3)$. But hold on a second! Before we declare victory, we need to make sure we can't factor anything further. This is a crucial step because sometimes the expression inside the parentheses can be factored even more. Think of it like double-checking your work to make sure you haven't missed anything.

To check for further factoring, we'll focus on the expression inside the parentheses: $(7 w^4 - 5 y^3 z^3)$. We need to see if there are any common factors between $7 w^4$ and $-5 y^3 z^3$.

Let's start by looking at the coefficients: 7 and -5. These numbers are both prime, and they don't share any common factors other than 1. So, we can't factor out any numerical common factors.

Now, let's look at the variables. We have $w^4$ in the first term and $y^3 z^3$ in the second term. Notice that there are no common variables between the two terms. The first term has w, while the second term has y and z. Since they don't share any variables, we can't factor out any variable common factors either.

So, what does this mean? It means that the expression $(7 w^4 - 5 y^3 z^3)$ cannot be factored further. There are no common factors between the terms, so we've taken the factoring as far as we can go.

This step is super important because it ensures that we've completely factored the expression. If we skipped this step, we might leave the expression in a partially factored state, which wouldn't be as simplified as possible. Always double-check to see if you can factor further, especially after you've factored out the GCF. It's like making sure you've tightened all the screws on a piece of furniture – you want to make sure everything is secure and in its final form.

Final Answer

Alright guys, we've reached the finish line! We started with the expression $21 w^6 y^4 z^5-15 w^2 y^7 z^8$, and we've gone through the process of factoring it step by step. We identified the Greatest Common Factor (GCF), factored it out, and then checked to see if we could factor further. And guess what? We've successfully factored the expression completely!

The final factored form of $21 w^6 y^4 z^5-15 w^2 y^7 z^8$ is:

3w2y4z5(7w4−5y3z3)3 w^2 y^4 z^5 (7 w^4 - 5 y^3 z^3)

This is our final answer. We've broken down the original expression into its simplest factors. We have the GCF, $3 w^2 y^4 z^5$, multiplied by the expression $(7 w^4 - 5 y^3 z^3)$, which cannot be factored further. This is the most simplified form of our original expression.

So, what have we learned today? We've learned how to factor expressions by identifying and factoring out the GCF. We've also learned the importance of checking for further factoring to ensure we've simplified the expression completely. Factoring is a fundamental skill in algebra, and it's something you'll use again and again in your math journey. By mastering these steps, you'll be well-equipped to tackle more complex factoring problems in the future.

Remember, factoring is like solving a puzzle. It might seem challenging at first, but with practice and a systematic approach, you can become a factoring pro! Keep practicing, and you'll find that factoring becomes second nature. And who knows, you might even start to enjoy it! So, until next time, keep factoring and keep learning!