Factoring (4x+3)(5x−8)+(4x+7)(4x+3) A Step-by-Step Guide

In the realm of mathematics, factoring expressions is a fundamental skill that unlocks the door to solving equations, simplifying complex formulas, and gaining a deeper understanding of algebraic relationships. When faced with an expression like (4x+3)(5x−8)+(4x+7)(4x+3), the task of factoring might seem daunting at first glance. But fear not, guys! With a systematic approach and a clear understanding of factoring techniques, we can break down this expression into its constituent factors and reveal its underlying structure. So, let's dive into the world of factoring and conquer this mathematical challenge together!

Understanding the Importance of Factoring

Before we embark on the journey of factoring the given expression, let's take a moment to appreciate the significance of factoring in mathematics. Factoring, in essence, is the reverse process of expansion. While expansion involves multiplying expressions to obtain a more complex form, factoring involves breaking down an expression into its simpler factors. This process is invaluable for several reasons:

• Solving Equations: Factoring plays a pivotal role in solving algebraic equations. When an equation is factored, it can be set equal to zero, and the individual factors can be solved independently, leading to the solutions of the original equation.
• Simplifying Expressions: Factoring can simplify complex expressions by identifying common factors and extracting them, resulting in a more concise and manageable form.
• Identifying Patterns: Factoring often reveals hidden patterns and relationships within expressions, providing insights into their mathematical nature.
• Graphing Functions: Factoring is instrumental in determining the x-intercepts (roots) of a function, which are crucial for sketching its graph.

With the importance of factoring firmly in mind, let's now turn our attention to the task at hand: factoring the expression (4x+3)(5x−8)+(4x+7)(4x+3).

Identifying Common Factors: The Key to Factoring

The first step in factoring any expression is to identify common factors. A common factor is an expression that appears in multiple terms of the given expression. In our case, the expression (4x+3) appears in both terms of the expression (4x+3)(5x−8)+(4x+7)(4x+3). This is a crucial observation, as it paves the way for factoring by grouping.

Factoring by grouping is a technique that involves identifying a common factor among two or more terms and extracting it. In our expression, we can see that (4x+3) is a common factor in both terms. So, let's factor it out:

(4x+3)(5x−8)+(4x+7)(4x+3) = (4x+3) [(5x−8) + (4x+7)]

Notice how we've extracted the common factor (4x+3) and placed the remaining expressions within brackets. This is the essence of factoring by grouping. Now, let's simplify the expression within the brackets.

Simplifying the Expression: Combining Like Terms

Inside the brackets, we have the expression (5x−8) + (4x+7). To simplify this, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, 5x and 4x are like terms, and -8 and 7 are like terms. Let's combine them:

(5x−8) + (4x+7) = 5x + 4x − 8 + 7 = 9x − 1

Now, we can substitute this simplified expression back into our factored expression:

(4x+3) [(5x−8) + (4x+7)] = (4x+3) (9x − 1)

The Factored Form: Unveiling the Components

We've successfully factored the expression! The factored form of (4x+3)(5x−8)+(4x+7)(4x+3) is (4x+3)(9x − 1). This factored form reveals the expression's constituent factors: (4x+3) and (9x − 1). These factors are the building blocks of the original expression, and understanding them provides valuable insights into its mathematical behavior.

By factoring the expression, we've transformed it from a sum of products into a product of two factors. This transformation can be immensely useful in various mathematical contexts, such as solving equations, simplifying expressions, and analyzing functions.

Expanding the Factored Form: Verifying Our Result

To ensure that our factoring is correct, we can expand the factored form and verify that it matches the original expression. Expansion involves multiplying the factors together to obtain the original expression. Let's expand (4x+3)(9x − 1) using the distributive property:

(4x+3)(9x − 1) = 4x(9x − 1) + 3(9x − 1) = 36x² − 4x + 27x − 3 = 36x² + 23x − 3

Now, let's expand the original expression (4x+3)(5x−8)+(4x+7)(4x+3):

(4x+3)(5x−8)+(4x+7)(4x+3) = (20x² − 32x + 15x − 24) + (16x² + 12x + 28x + 21) = 20x² − 17x − 24 + 16x² + 40x + 21 = 36x² + 23x − 3

As we can see, the expanded form of the factored expression matches the expanded form of the original expression. This confirms that our factoring is indeed correct. We've successfully factored the expression (4x+3)(5x−8)+(4x+7)(4x+3) into its constituent factors (4x+3) and (9x − 1).

Applications of Factoring: Beyond the Basics

Factoring is not just an abstract mathematical exercise; it has numerous applications in various fields, including:

• Engineering: Factoring is used in structural analysis, circuit design, and control systems.
• Physics: Factoring appears in mechanics, electromagnetism, and quantum mechanics.
• Computer Science: Factoring is used in cryptography, data compression, and algorithm design.
• Economics: Factoring is used in financial modeling, optimization problems, and game theory.

The ability to factor expressions is a valuable asset in these fields, as it allows professionals to simplify complex problems, identify patterns, and develop effective solutions.

Conclusion: Mastering the Art of Factoring

Factoring the expression (4x+3)(5x−8)+(4x+7)(4x+3) might have seemed like a daunting task at first, but by following a systematic approach, we've successfully broken it down into its constituent factors. We've learned the importance of identifying common factors, simplifying expressions, and verifying our results through expansion. Factoring is a fundamental skill in mathematics, and mastering it opens doors to solving equations, simplifying expressions, and gaining a deeper understanding of mathematical relationships.

So, keep practicing, guys! The more you factor, the more proficient you'll become. And remember, the world of mathematics is full of fascinating challenges waiting to be conquered. Embrace the challenge, and let the power of factoring be your guide!