Factorizing 8x²y² + 4x² - 12x²y A Step-by-Step Guide

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Introduction

In this article, we will delve into the process of factorizing the algebraic expression 8x²y² + 4x² - 12x²y. Factorization is a fundamental concept in algebra, allowing us to simplify complex expressions and solve equations more easily. We will break down the expression step by step, identify common factors, and ultimately express the given expression in its factored form. Understanding factorization is crucial for various mathematical operations, including solving quadratic equations, simplifying rational expressions, and tackling more advanced algebraic problems.

Understanding Factorization

Before we dive into the specific expression, let's briefly discuss what factorization entails. Factorization is the process of expressing a number or an algebraic expression as a product of its factors. In simpler terms, it's like reversing the process of expansion or distribution. For instance, if we expand the expression 2(x + 3), we get 2x + 6. Conversely, factorizing 2x + 6 would give us 2(x + 3). The goal of factorization is to break down a complex expression into simpler components that, when multiplied together, yield the original expression.

In the context of algebraic expressions, factors can be numbers, variables, or even other algebraic expressions. The ability to factorize expressions is a cornerstone of algebraic manipulation and is essential for simplifying complex equations and solving problems efficiently. There are various techniques for factorization, including finding the greatest common factor (GCF), using identities, and grouping terms. Each technique is suited for different types of expressions, and mastering these techniques is crucial for success in algebra.

Step-by-Step Factorization of 8x²y² + 4x² - 12x²y

Now, let's tackle the expression 8x²y² + 4x² - 12x²y step by step. Our aim is to factorize this expression into its simplest form.

1. Identifying the Greatest Common Factor (GCF)

The first step in factorization is to identify the greatest common factor (GCF) among the terms. The GCF is the largest factor that divides all the terms in the expression. In our expression, 8x²y² + 4x² - 12x²y, the terms are 8x²y², 4x², and -12x²y.

Let's break down each term to find the GCF:

  • 8x²y² = 2 × 2 × 2 × x × x × y × y
  • 4x² = 2 × 2 × x × x
  • -12x²y = -1 × 2 × 2 × 3 × x × x × y

By examining the prime factorization of each term, we can identify the common factors. The common factors are 2 × 2 × x × x, which equals 4x². Therefore, the GCF of the expression is 4x². Identifying the GCF is a critical step because it allows us to simplify the expression significantly before proceeding with other factorization techniques. Failing to identify the GCF can lead to more complicated steps later on, so it’s essential to start with this step.

2. Factoring out the GCF

Once we have identified the GCF, which is 4x², the next step is to factor it out from the expression. This involves dividing each term in the expression by the GCF and writing the expression as the product of the GCF and the resulting quotient.

Dividing each term by 4x²:

  • (8x²y²) / (4x²) = 2y²
  • (4x²) / (4x²) = 1
  • (-12x²y) / (4x²) = -3y

Now, we can rewrite the original expression by factoring out 4x²:

8x²y² + 4x² - 12x²y = 4x²(2y² + 1 - 3y)

This step is crucial as it simplifies the original expression into a product of a monomial (4x²) and a trinomial (2y² + 1 - 3y). Factoring out the GCF not only makes the expression more manageable but also reveals the underlying structure, which is essential for further factorization if needed. The expression inside the parentheses, 2y² + 1 - 3y, is a quadratic trinomial that we will analyze in the next step to see if it can be factored further.

3. Rearranging and Factoring the Trinomial

After factoring out the GCF, we are left with the trinomial 2y² + 1 - 3y inside the parentheses. To make it easier to factor, we should first rearrange the terms in descending order of the variable's exponent. This means rewriting the trinomial as 2y² - 3y + 1. Arranging the terms in this order helps in recognizing potential factoring patterns and applying the appropriate techniques.

Now, we need to factorize the quadratic trinomial 2y² - 3y + 1. There are several methods to factorize a quadratic trinomial, such as trial and error, using the quadratic formula, or factoring by grouping. In this case, we will use the factoring by grouping method. Factoring by grouping involves finding two numbers that multiply to the product of the leading coefficient (2) and the constant term (1), which is 2, and add up to the middle coefficient (-3). These two numbers are -2 and -1.

We can rewrite the middle term (-3y) as the sum of -2y and -1y:

2y² - 3y + 1 = 2y² - 2y - y + 1

Next, we group the terms in pairs:

(2y² - 2y) + (-y + 1)

Now, we factor out the GCF from each group:

2y(y - 1) - 1(y - 1)

We can see that (y - 1) is a common factor in both terms. Factoring it out, we get:

(2y - 1)(y - 1)

So, the factored form of the trinomial 2y² - 3y + 1 is (2y - 1)(y - 1). This step completes the factorization of the trinomial part of our original expression, and it's a crucial step in achieving the fully factored form.

4. Final Factored Form

Now that we have factored the trinomial, we can write the complete factored form of the original expression. Recall that we factored out 4x² from the original expression and then factored the trinomial 2y² - 3y + 1 into (2y - 1)(y - 1). Combining these results, we get the final factored form:

8x²y² + 4x² - 12x²y = 4x²(2y - 1)(y - 1)

This is the fully factored form of the given expression. It represents the original expression as a product of its factors, which are 4x², (2y - 1), and (y - 1). This final form is significantly simpler and easier to work with compared to the original expression. Verifying the result by expanding the factored form will give the original expression, ensuring the factorization is correct. This process demonstrates the power of factorization in simplifying complex algebraic expressions and making them more manageable for various mathematical operations.

Alternative Representation and Discussion

The factored form 4x²(2y - 1)(y - 1) is the most common and simplified representation. However, there can be alternative ways to express the factors, although they are essentially equivalent. For example, the order of the factors does not matter due to the commutative property of multiplication. Therefore, expressions like (y - 1)(2y - 1)4x² or (2y - 1)4x²(y - 1) are equally valid.

It's important to understand that factorization is not always unique in terms of the arrangement of factors, but the factors themselves must be the same. The significance of factorization lies in its ability to simplify complex expressions, which is invaluable in solving equations, simplifying rational expressions, and performing calculus operations. Recognizing patterns and applying appropriate factorization techniques are essential skills in algebra and beyond. In the context of problem-solving, a factored expression can often reveal key insights and lead to more efficient solutions.

Conclusion

In this article, we have successfully factorized the expression 8x²y² + 4x² - 12x²y into its factored form 4x²(2y - 1)(y - 1). We achieved this by first identifying and factoring out the greatest common factor (GCF), which was 4x². Then, we rearranged and factored the resulting trinomial 2y² - 3y + 1 using the factoring by grouping method. This step-by-step approach highlights the importance of understanding and applying different factorization techniques.

Factorization is a fundamental skill in algebra with wide-ranging applications. Mastering this skill allows for simplification of complex expressions, efficient solving of equations, and a deeper understanding of algebraic structures. The ability to recognize and apply factorization techniques is crucial for success in mathematics and related fields. By practicing and understanding the underlying principles, one can become proficient in factorizing various types of algebraic expressions, making complex problems more manageable and accessible. The example we worked through illustrates a systematic approach to factorization, emphasizing the importance of breaking down the problem into smaller, more manageable steps. This approach can be applied to a wide range of factorization problems, making it a valuable tool in any mathematician's toolkit.