Factoring Completely: $3w^2 - 3u^4w^2$ - A Step-by-Step Guide

In this comprehensive guide, we will walk through the process of factoring the algebraic expression $3w^2 - 3u^4w^2$ completely. Factoring is a fundamental skill in algebra, allowing us to simplify expressions, solve equations, and gain a deeper understanding of mathematical relationships. Whether you're a student looking to improve your algebra skills or simply curious about mathematical problem-solving, this article will provide a clear and detailed explanation. Let's dive in and break down this problem step by step. Understanding how to factor expressions like this is crucial for various mathematical applications, so mastering this technique will undoubtedly benefit you in the long run.

1. Identifying the Greatest Common Factor (GCF)

To begin factoring the expression $3w^2 - 3u^4w^2$ completely, the first step is to identify the greatest common factor (GCF). The GCF is the largest factor that divides evenly into all terms of the expression. In this case, we need to look at both the coefficients (the numerical parts) and the variables (the literal parts) of the terms.

Examining the Coefficients

First, let's consider the coefficients: 3 and -3. The greatest common factor of 3 and -3 is 3. This is because 3 is the largest number that divides both 3 and -3 without leaving a remainder. It's important to consider both positive and negative factors, but in this case, we'll focus on the positive GCF for simplicity and convention. Factoring out the GCF simplifies the expression and makes further factoring steps easier to manage. This initial step is vital for breaking down complex expressions into more manageable parts.

Analyzing the Variables

Next, we need to look at the variables. We have $w^2$ in both terms, and $u^4$ only in the second term. The greatest common factor for the variable part is the lowest power of the common variable. Here, $w^2$ is present in both terms, so it is part of the GCF. The variable $u$ is only in one term, so it is not included in the GCF. Therefore, the variable part of our GCF is $w^2$. Recognizing common variables and their lowest powers is key to correctly identifying the GCF. This ensures that the factored expression is accurate and simplified to its fullest extent.

Combining Coefficients and Variables

Combining the greatest common factor of the coefficients (3) and the greatest common factor of the variables ($w^2$), we find that the overall GCF for the expression $3w^2 - 3u^4w^2$ is $3w^2$. Identifying the GCF correctly is the cornerstone of effective factoring, paving the way for the subsequent steps in the process. Now that we've found the GCF, we can proceed to factor it out of the original expression. This step will significantly simplify the expression and make it easier to recognize any remaining factors.

2. Factoring out the GCF

Now that we've identified the greatest common factor (GCF) as $3w^2$, the next step is to factor it out of the expression $3w^2 - 3u^4w^2$. Factoring out the GCF involves dividing each term in the original expression by the GCF and writing the result in a factored form. This process effectively reverses the distributive property and allows us to rewrite the expression in a more simplified manner.

Dividing Each Term by the GCF

We start by dividing each term of the expression by $3w^2$:

• For the first term, $3w^2$, we have $3w^2 / 3w^2 = 1$.
• For the second term, $-3u^4w^2$, we have $-3u^4w^2 / 3w^2 = -u^4$.

When dividing terms with exponents, we subtract the exponents of like variables. In this case, $w^2$ divided by $w^2$ cancels out, leaving us with 1 in the first term and $-u^4$ in the second term. This division process is crucial for accurately factoring out the GCF and simplifying the expression. By carefully dividing each term, we ensure that we are maintaining the equality of the expression while transforming its form.

Writing the Factored Expression

Now, we rewrite the original expression by factoring out the GCF. We place the GCF, $3w^2$, outside a set of parentheses, and inside the parentheses, we write the results of our division:

$3w^2(1 - u^4)$

This is the factored form of the expression after factoring out the GCF. We have successfully taken the common factor out of both terms, resulting in a more simplified expression. However, the factoring process may not be complete at this stage. It's important to check the expression inside the parentheses to see if it can be factored further. In this case, the expression $(1 - u^4)$ is a difference of squares, which can be factored again. Recognizing these patterns and continuing the factoring process is essential for completely factoring the original expression. This factored form is a stepping stone towards further simplification and ultimately solving the problem.

3. Recognizing and Factoring the Difference of Squares

After factoring out the GCF, we have the expression $3w^2(1 - u^4)$. Now, we need to examine the expression inside the parentheses, $(1 - u^4)$, to see if it can be factored further. We can recognize that $(1 - u^4)$ is a difference of squares. The difference of squares is a special pattern in algebra where we have the form $a^2 - b^2$, which can be factored as $(a - b)(a + b)$. Recognizing this pattern is crucial for factoring expressions completely.

Identifying the Squares

To apply the difference of squares pattern, we need to identify the terms that are perfect squares. In the expression $(1 - u^4)$, we can rewrite 1 as $1^2$ and $u^4$ as $(u^2)^2$. Thus, we have:

$1^2 - (u^2)^2$

Now it is clear that we have a difference of squares, where $a = 1$ and $b = u^2$. Being able to identify perfect squares and rewrite terms in this form is a key step in applying the difference of squares pattern. This allows us to break down the expression into its factored components.

Applying the Difference of Squares Formula

Using the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$, we can factor $(1 - u^4)$ as follows:

$(1 - u^2)(1 + u^2)$

We have now factored $(1 - u^4)$ into $(1 - u^2)(1 + u^2)$. However, we are not done yet! Notice that $(1 - u^2)$ is itself a difference of squares. It's essential to continue factoring until no further factoring is possible. This ensures that the expression is completely factored.

Factoring Again: Another Difference of Squares

We can apply the difference of squares pattern again to $(1 - u^2)$. Here, we have $1^2 - (u)^2$, so $a = 1$ and $b = u$. Factoring this, we get:

$(1 - u)(1 + u)$

The expression $(1 - u^2)$ factors into $(1 - u)(1 + u)$. Now, we have factored this part completely. The term $(1 + u^2)$ cannot be factored further using real numbers, as it is a sum of squares, which does not have a simple factoring pattern like the difference of squares. Understanding when to stop factoring is just as important as knowing how to factor. Once we've exhausted all possible factoring techniques, we can be confident that the expression is fully simplified.

4. Writing the Completely Factored Expression

Now that we have factored all parts of the expression as much as possible, we can write the completely factored form. We started with the expression $3w^2 - 3u^4w^2$, factored out the GCF to get $3w^2(1 - u^4)$, and then factored the difference of squares $(1 - u^4)$ into $(1 - u^2)(1 + u^2)$, and further factored $(1 - u^2)$ into $(1 - u)(1 + u)$. Combining all these steps, we can now write the final factored expression.

Combining the Factors

We have the following factors:

• $3w^2$ (the GCF)
• $(1 - u)(1 + u)$ (from factoring $1 - u^2$)
• $(1 + u^2)$ (from the earlier difference of squares)

Putting these together, the completely factored expression is:

$3w^2(1 - u)(1 + u)(1 + u^2)$

This is the completely factored form of the original expression. We have taken the initial expression and broken it down into its simplest factors. This is an essential skill in algebra for simplifying expressions, solving equations, and understanding mathematical relationships.

Importance of Complete Factorization

It's important to ensure that an expression is factored completely. If we had stopped at an earlier stage, such as $3w^2(1 - u^4)$, we would not have fully simplified the expression. Complete factorization allows us to see all the factors and simplifies further calculations or analysis involving the expression. Complete factorization also helps in identifying roots of polynomial equations and understanding the behavior of functions. The factored form can reveal important information that is not readily apparent in the original expression. Mastering the techniques of factoring, including recognizing patterns like the difference of squares, is crucial for success in algebra and beyond.

Conclusion

In this step-by-step guide, we successfully factored the expression $3w^2 - 3u^4w^2$ completely. We began by identifying and factoring out the greatest common factor (GCF), which was $3w^2$. This simplified the expression to $3w^2(1 - u^4)$. Then, we recognized the difference of squares pattern in $(1 - u^4)$ and factored it into $(1 - u^2)(1 + u^2)$. We further factored $(1 - u^2)$ as another difference of squares, resulting in $(1 - u)(1 + u)$. Finally, we combined all the factors to obtain the completely factored expression: $3w^2(1 - u)(1 + u)(1 + u^2)$.

Factoring is a crucial skill in algebra and mathematics in general. It simplifies expressions, helps in solving equations, and provides insights into the structure of mathematical relationships. Mastering factoring techniques such as identifying the GCF and recognizing special patterns like the difference of squares is essential for success in algebra and beyond. By following a systematic approach and carefully examining each part of an expression, you can factor complex expressions completely and accurately.

To further enhance your understanding of factoring and related algebraic concepts, you may find the resources available at Khan Academy's Algebra Section particularly helpful.