Solving (a-b-1)(a² + B² + 1 + Ab + A - B): A Step-by-Step Guide

by TextBrain Team 64 views

Hey guys! Let's dive into this math problem together. We've got a rather interesting expression to tackle: (a-b-1)(a² + b² + 1 + ab + a - b). It looks intimidating, but don't worry! We'll break it down step by step, making it super easy to understand. Our goal is to simplify this expression, and we'll do that by carefully expanding and combining like terms. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's first understand what we're dealing with. The problem presents us with a product of two expressions. The first expression is (a - b - 1), a simple trinomial. The second expression is (a² + b² + 1 + ab + a - b), a more complex polynomial. To solve this, we need to multiply each term in the first expression by each term in the second expression and then simplify the result.

Why is this important? Well, these types of algebraic manipulations are fundamental in various fields, including engineering, computer science, and even economics. Mastering this skill helps in solving equations, understanding functions, and building mathematical models. It's like learning the ABCs of mathematics – essential for further exploration.

The key here is to be methodical and organized. We'll go through each multiplication step by step, ensuring we don't miss any terms. Think of it like building a house; each brick (or term) needs to be placed correctly to ensure a solid structure (or simplified expression).

Now, let’s look at the expressions individually. (a - b - 1) has three terms: 'a', '-b', and '-1'. The second expression, (a² + b² + 1 + ab + a - b), has six terms: 'a²', 'b²', '1', 'ab', 'a', and '-b'. This means we'll have a total of 3 * 6 = 18 multiplications to perform. Sounds like a lot, right? But don’t fret! We'll take it one step at a time.

Breaking Down the Complexity

One effective strategy when dealing with complex expressions is to break them down into smaller, more manageable parts. Instead of trying to multiply everything at once, we'll distribute each term of the first expression across the second expression individually. This approach reduces the chance of making errors and helps keep our work organized.

For instance, we’ll first multiply 'a' from (a - b - 1) with each term in (a² + b² + 1 + ab + a - b). Then, we’ll do the same with '-b', and finally with '-1'. By breaking the problem into these three smaller sub-problems, we simplify the task significantly.

Another important aspect to consider is the order of operations. Remember the good old PEMDAS/BODMAS rule? (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). While there are no explicit parentheses to simplify within the expressions, we'll be focusing on multiplication first, followed by combining like terms (which involves addition and subtraction).

So, with a clear strategy and understanding of the problem, we're well-equipped to tackle this algebraic challenge. Let's roll up our sleeves and start multiplying!

Step-by-Step Solution

Okay, let's get into the heart of the problem! We'll take this expression (a-b-1)(a² + b² + 1 + ab + a - b) and break it down systematically. Remember, we're going to multiply each term in the first expression by each term in the second expression. It's a bit like a mathematical dance, where each term gets its turn in the spotlight.

Step 1: Distribute 'a'

First, we'll multiply 'a' from the first expression (a - b - 1) with every term in the second expression (a² + b² + 1 + ab + a - b). This gives us:

a * (a² + b² + 1 + ab + a - b) = a³ + ab² + a + a²b + a² - ab

See? It's not so scary when we take it term by term. We've just expanded the first part of our expression. Now, let's move on to the next term.

Step 2: Distribute '-b'

Next, we'll multiply '-b' from the first expression with each term in the second expression. This is where we need to be extra careful with our signs! Here we go:

-b * (a² + b² + 1 + ab + a - b) = -a²b - b³ - b - ab² - ab + b²

Notice how each term has '-b' multiplied with it, changing the signs accordingly. We're making progress! One more term to go.

Step 3: Distribute '-1'

Finally, we'll multiply '-1' with each term in the second expression. This step is a bit simpler, as it just involves changing the sign of each term:

-1 * (a² + b² + 1 + ab + a - b) = -a² - b² - 1 - ab - a + b

We've now completed all the multiplications. Give yourself a pat on the back – the hard part is over!

Step 4: Combine All the Terms

Now comes the exciting part: putting it all together! We'll combine the results from the previous steps. This means we'll add the expressions we got in steps 1, 2, and 3:

(a³ + ab² + a + a²b + a² - ab) + (-a²b - b³ - b - ab² - ab + b²) + (-a² - b² - 1 - ab - a + b)

It looks like a jumbled mess right now, but don't worry. We're about to bring some order to this chaos!

Step 5: Simplify by Combining Like Terms

This is where we hunt for terms that are similar and can be combined. Like terms are those that have the same variables raised to the same powers. For example, 'a²b' and '-a²b' are like terms, while 'a²' and 'ab' are not.

Let's go through the expression and combine those like terms:

  • a³: There's only one 'a³' term, so it stays as it is.
  • ab²: We have 'ab²' and '-ab²', which cancel each other out (ab² - ab² = 0).
  • a²b: We have 'a²b' and '-a²b', which also cancel each other out (a²b - a²b = 0).
  • a²: We have 'a²' and '-a²', which cancel each other out (a² - a² = 0).
  • b²: We have 'b²' and '-b²', which cancel each other out (b² - b² = 0).
  • b³: There's only one '-b³' term, so it stays as it is.
  • a: We have 'a' and '-a', which cancel each other out (a - a = 0).
  • ab: We have '-ab', '-ab', and '-ab', which combine to -3ab.
  • b: We have '-b' and '+b', which cancel each other out (-b + b = 0).
  • Constants: We have '+1' and '-1', so we are left with '-1'.

Step 6: Write the Simplified Expression

After canceling and combining all the like terms, we're left with a much simpler expression:

a³ - b³ - 3ab - 1

And there you have it! The simplified form of the original expression is a³ - b³ - 3ab - 1. Wasn't that satisfying? We took a complex expression and, through careful steps, reduced it to its simplest form.

Common Mistakes to Avoid

Guys, when tackling problems like this, it's super easy to slip up if you're not careful. Let's chat about some common mistakes so you can dodge them like a pro.

Sign Errors

  • Why it happens: Sign errors are like the ninjas of algebra – sneaky and silent! They often occur during the distribution of negative terms. For instance, forgetting to multiply the negative sign when distributing '-b' across the second expression. It’s a classic blunder that can throw off your entire solution.
  • How to avoid it: Double-check each term after distributing a negative sign. Make sure every sign is flipped correctly. It’s like proofreading your work, but for math. Pay extra attention when you see those minus signs lurking around!

Incorrect Distribution

  • Why it happens: This is when you forget to multiply a term by every single term within the parentheses. It’s like inviting some guests to a party but accidentally missing a few – not cool! This usually happens when we try to rush through the process or lose focus amidst all the terms.
  • How to avoid it: Use a systematic approach. Go term by term, ensuring you've hit every combination. You can even draw little arrows connecting the terms you’re multiplying, just to keep track. Think of it as a mathematical checklist!

Combining Unlike Terms

  • Why it happens: This is a mix-up where you try to add or subtract terms that aren't