Factor 4y³ - 100y: A Step-by-Step Guide

by Luna Greco 40 views

Hey guys! Today, we're going to dive into the world of factoring, specifically focusing on the expression 4y³ - 100y. Factoring is like the reverse of expanding; instead of multiplying terms together, we're breaking them down into their simplest factors. This is super useful in algebra for solving equations, simplifying expressions, and all sorts of other mathematical wizardry. So, let's get started and break down this problem step by step!

1. Understanding the Basics of Factoring

Before we jump into the specifics of 4y³ - 100y, let's quickly recap what factoring actually means. In essence, factoring is the process of finding the expressions that, when multiplied together, give you the original expression. Think of it like this: if you have the number 12, you can factor it into 3 x 4, or 2 x 6, or even 2 x 2 x 3. All of these are factors of 12. We aim to do the same thing with algebraic expressions, but instead of numbers, we're dealing with variables and constants.

Now, when it comes to factoring polynomials (expressions with multiple terms), there are a few common techniques we often use. The most basic one, and the one we'll use primarily today, is finding the greatest common factor (GCF). The GCF is the largest factor that divides evenly into all terms of the polynomial. Once we identify the GCF, we can factor it out, which simplifies the expression. There are also other factoring methods like recognizing differences of squares, perfect square trinomials, and factoring by grouping, but for 4y³ - 100y, the GCF method will be our star player. So, keep this concept of the greatest common factor in mind as we tackle our problem.

2. Identifying the Greatest Common Factor (GCF)

The first crucial step in factoring 4y³ - 100y is to identify the greatest common factor (GCF). Remember, the GCF is the largest factor that divides evenly into both terms of our expression. Looking at 4y³ and 100y, we need to consider both the numerical coefficients (4 and 100) and the variable parts (y³ and y).

Let's start with the numerical coefficients. What's the largest number that divides evenly into both 4 and 100? Well, 4 goes into 4 once, and 4 goes into 100 twenty-five times. So, 4 is a common factor. Is it the greatest common factor? Yes, because no number larger than 4 can divide evenly into 4. So, our numerical GCF is 4.

Now, let's look at the variable parts. We have y³ (which means y * y * y) and y (which means y). What's the largest power of y that is common to both terms? Both terms have at least one y, so y is a common factor. However, y³ has y multiplied three times, while the second term only has y to the power of 1. Therefore, the greatest common variable factor is simply y (or y¹ if you want to be precise). Combining the numerical and variable GCFs, we find that the greatest common factor of 4y³ and 100y is 4y. This is the key that will unlock our factoring process, so make sure you're comfortable with how we found it. This GCF will be pulled out in the next step, making the expression simpler and easier to handle.

3. Factoring Out the GCF from 4y³ - 100y

Now that we've identified the greatest common factor (GCF) as 4y, we can proceed to factor it out from our expression, 4y³ - 100y. Factoring out the GCF involves dividing each term in the expression by the GCF and then writing the expression as a product of the GCF and the resulting quotient. Think of it like reverse distribution.

Here's how it works: we take each term in 4y³ - 100y and divide it by 4y. So, we have 4y³ / 4y and -100y / 4y. Let's do these divisions separately. For 4y³ / 4y, the 4s cancel out, and we're left with y³ / y. Remember the rules of exponents: when dividing like bases, you subtract the exponents. So, y³ / y is the same as y^(3-1), which simplifies to y². Moving on to the second term, -100y / 4y, we divide -100 by 4, which gives us -25. The ys cancel each other out (y / y = 1), so we're left with -25.

Now, we can rewrite our original expression, 4y³ - 100y, as the product of the GCF and the results of our divisions. This looks like 4y(y² - 25). What we've done here is essentially "pulled out" the 4y from both terms, leaving us with the expression inside the parentheses. This is a huge step forward in factoring because we've simplified the original expression. However, we're not quite done yet! The expression inside the parentheses, (y² - 25), looks familiar, and there's another factoring technique we can apply to it.

4. Recognizing and Factoring the Difference of Squares

Okay, we've successfully factored out the greatest common factor (GCF) from 4y³ - 100y, and we're now at 4y(y² - 25). Take a good look at the expression inside the parentheses: (y² - 25). Does it look familiar to you? It should! This is a classic example of what we call the difference of squares. The difference of squares pattern is one of those really useful shortcuts in factoring, and recognizing it can save you a lot of time and effort.

The general form of a difference of squares is a² - b², where 'a' and 'b' can be any algebraic terms. The magic of the difference of squares is that it always factors into a specific pattern: (a + b)(a - b). This means if you can identify an expression in the form of something squared minus something else squared, you can immediately factor it into the sum and difference of those somethings. Now, let's see how this applies to our expression, (y² - 25).

Can we rewrite y² - 25 in the form a² - b²? Absolutely! y² is clearly something squared (y squared). And 25 is also something squared – it's 5 squared (5² = 25). So, we can think of y² - 25 as y² - 5². Now, we have a perfect match for the difference of squares pattern, where 'a' is y and 'b' is 5. Applying our formula, (a² - b²) = (a + b)(a - b), we can factor y² - 25 into (y + 5)(y - 5). We are getting closer to fully factoring the expression!

5. Completing the Factoring Process

We've done the hard work of identifying the greatest common factor (GCF) and recognizing the difference of squares. Now, it's time to put it all together and complete the factoring process for 4y³ - 100y. Remember, we started by factoring out the GCF, 4y, which gave us 4y(y² - 25). Then, we recognized that (y² - 25) is a difference of squares and factored it into (y + 5)(y - 5).

So, to get the fully factored form of our original expression, we simply combine these two steps. We replace (y² - 25) with its factored form, (y + 5)(y - 5), in our expression. This gives us 4y(y + 5)(y - 5). And that's it! We've successfully factored 4y³ - 100y completely.

Let's take a moment to appreciate what we've accomplished. We started with a cubic expression (an expression with a term raised to the power of 3) and broke it down into a product of simpler factors. The factored form, 4y(y + 5)(y - 5), is equivalent to the original expression, but it's much easier to work with in many situations. For example, if you needed to solve the equation 4y³ - 100y = 0, the factored form makes it a breeze. You can simply set each factor equal to zero (4y = 0, y + 5 = 0, y - 5 = 0) and solve for y, giving you the solutions y = 0, y = -5, and y = 5. Factoring is truly a powerful tool in algebra.

6. Final Answer and Review

Alright, let's wrap things up and make sure we're crystal clear on the final answer. We started with the expression 4y³ - 100y and went through a step-by-step process of factoring. We identified the greatest common factor (GCF) as 4y, factored it out, recognized the difference of squares pattern, and factored that as well. So, after all our hard work, the completely factored form of 4y³ - 100y is:

4y(y + 5)(y - 5)

That's it! This is our final answer, and it represents the original expression broken down into its simplest factors. Now, let's quickly recap the key steps we took to get here:

  1. Identify the GCF: We found that 4y was the greatest common factor of 4y³ and 100y.
  2. Factor out the GCF: We rewrote the expression as 4y(y² - 25).
  3. Recognize the difference of squares: We identified (y² - 25) as a difference of squares (y² - 5²).
  4. Factor the difference of squares: We factored (y² - 25) into (y + 5)(y - 5).
  5. Combine the factors: We put it all together to get the final factored form: 4y(y + 5)(y - 5).

By following these steps, you can tackle many factoring problems with confidence. Remember, practice makes perfect, so the more you factor, the better you'll become at recognizing patterns and applying the right techniques. Great job working through this problem with me, guys! Keep up the excellent work, and happy factoring!