Expanding And Simplifying (t-2)(t+5)(t-4): A Step-by-Step Guide

Let's dive into the world of algebra and tackle the problem of expanding and simplifying the expression (t-2)(t+5)(t-4). This is a common type of problem you'll encounter in algebra, and mastering it will build a solid foundation for more complex mathematical concepts. We'll break it down step by step, making it easy to follow along, even if you're just starting out with algebra. So, grab your pencil and paper, and let's get started!

Understanding the Basics

Before we jump into the expansion, it’s essential to understand what we mean by "expanding" and "simplifying". Expanding refers to multiplying out the terms within the parentheses. Simplifying, on the other hand, means combining like terms to write the expression in its most concise form. The goal is to eliminate parentheses and combine similar terms (like terms with the same variable and exponent) to arrive at a neat, manageable expression. Understanding these core ideas sets the stage for a smooth and efficient solution. When you see expressions like these, remember that it's all about distributing the multiplication and then tidying up the result by combining what can be combined.

Now, why is this important? Expanding and simplifying expressions is a fundamental skill in algebra, and it shows up everywhere. From solving equations to graphing functions, you'll constantly use these techniques. Think of it as learning the alphabet of mathematics – you need to know it to read and write more complex mathematical "sentences". This skill also boosts your problem-solving abilities. By breaking down a complex expression into simpler parts, you can better understand its structure and behavior. This makes it easier to manipulate the expression, solve equations, and even make predictions in real-world scenarios. For instance, in physics, you might use these skills to simplify equations describing motion or forces. In economics, you might use them to analyze cost and revenue functions. The applications are endless, making it a truly valuable skill to acquire.

Step 1: Expanding the First Two Factors

The first step in simplifying this expression is to expand the first two factors: (t-2) and (t+5). We'll use the distributive property, sometimes referred to as the FOIL method (First, Outer, Inner, Last), to multiply these two binomials.

Here’s how it works:

• First: Multiply the first terms in each binomial: t * t = t²
• Outer: Multiply the outer terms: t * 5 = 5t
• Inner: Multiply the inner terms: -2 * t = -2t
• Last: Multiply the last terms: -2 * 5 = -10

Now, let's combine these results:

t² + 5t - 2t - 10

Next, we simplify by combining the like terms (5t and -2t):

t² + 3t - 10

So, (t-2)(t+5) expands to t² + 3t - 10. This is an important intermediate result, as we'll use it in the next step.

Let's think about what we just did. We took two relatively simple expressions and combined them into a single, slightly more complex one. This is a common theme in algebra: breaking down a problem into smaller, more manageable parts. By focusing on each step individually, we reduce the chance of making mistakes and gain a clearer understanding of the overall process. It's like building a house brick by brick; each brick is small and manageable, but together, they form a sturdy structure.

Step 2: Multiplying the Result by the Third Factor

Now that we've simplified (t-2)(t+5) to t² + 3t - 10, we need to multiply this result by the remaining factor, (t-4). Again, we'll use the distributive property, but this time, we're multiplying a trinomial (three terms) by a binomial (two terms).

Here's how we do it. We will distribute each term in the binomial (t-4) across the entire trinomial (t² + 3t - 10).

First, multiply t by each term in the trinomial:

t * (t² + 3t - 10) = t³ + 3t² - 10t

Next, multiply -4 by each term in the trinomial:

-4 * (t² + 3t - 10) = -4t² - 12t + 40

Now, we add these two results together:

t³ + 3t² - 10t - 4t² - 12t + 40

Step 3: Combining Like Terms

The final step is to combine like terms to simplify the expression. Look for terms with the same variable and exponent.

In our expression, t³ + 3t² - 10t - 4t² - 12t + 40, we have the following like terms:

• t² terms: 3t² and -4t²
• t terms: -10t and -12t

Let's combine them:

• 3t² - 4t² = -t²
• -10t - 12t = -22t

Now, we rewrite the expression with the combined like terms:

t³ - t² - 22t + 40

This is the fully simplified form of the original expression. Notice how we've reduced it from a product of three factors to a single polynomial. This simplified form is often easier to work with in further calculations or analysis.

So, the fully simplified expression is t³ - t² - 22t + 40. This is the final answer. We started with a product of three binomials and ended up with a simplified polynomial. This process of expanding and simplifying is a fundamental skill in algebra and will come in handy in many different contexts.

To recap, we first expanded the first two factors using the distributive property (FOIL method). Then, we multiplied the result by the third factor, again using the distributive property. Finally, we combined like terms to arrive at the fully simplified expression. Remember to take your time, be careful with your signs, and double-check your work. With practice, you'll become more confident and efficient in expanding and simplifying algebraic expressions.

Conclusion

We've successfully expanded and simplified the expression (t-2)(t+5)(t-4) to get t³ - t² - 22t + 40. This exercise demonstrates the power of the distributive property and the importance of combining like terms. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic problems. Remember to practice regularly, and don't be afraid to break down complex problems into smaller, more manageable steps.

Keep practicing, and you'll find that these types of problems become second nature. The key is to understand the underlying principles and apply them consistently. With dedication and effort, you can conquer any algebraic challenge that comes your way.

For more information on algebraic expressions and simplification techniques, you can visit Khan Academy's Algebra Section. This is an external link.