Solving $(x^7 + X^9 + X^{12})^2$ A Step-by-Step Guide

Hey there, math enthusiasts! Let's dive into this interesting algebraic expression: $(x^7 + x^9 + x^{12})^2$. It looks a bit intimidating at first glance, but don't worry, we'll break it down step by step. This detailed guide will not only help you understand the expression but also equip you with the knowledge to tackle similar problems. Whether you're preparing for an exam or just curious about algebra, this is the perfect place to start. So, let's roll up our sleeves and get to work!

Understanding the Basics

Before we jump into the nitty-gritty, let’s make sure we’re all on the same page with the basics. When you see an expression like $(x^7 + x^9 + x^{12})^2$, it essentially means you're multiplying the expression inside the parenthesis by itself. Think of it as $(A)^2 = A * A$. So in our case:

$(x^7 + x^9 + x^{12})^2 = (x^7 + x^9 + x^{12}) * (x^7 + x^9 + x^{12})$

Now, the next key concept to remember is the distributive property. This property tells us how to multiply a sum by a sum. If we have something like $(a + b + c) * (d + e + f)$, we need to multiply each term in the first parenthesis by each term in the second parenthesis and then add them all up. It sounds like a lot, but we'll take it one step at a time. Let’s break down this process even further.

In our specific problem, we have three terms inside each parenthesis: $x^7$, $x^9$, and $x^{12}$. This means we will have 3 * 3 = 9 multiplications to perform. It’s crucial to keep things organized to avoid any mistakes. We'll go through each multiplication systematically.

Additionally, understanding the rules of exponents is vital. Remember that when you multiply terms with the same base, you add their exponents. For example, $x^a * x^b = x^{a+b}$. This rule will be our best friend as we work through this problem. Using exponents correctly is a core part of handling polynomial expressions, and mastering this will make your algebraic manipulations much smoother. Don’t forget this handy rule as we move forward!

Step-by-Step Expansion

Alright, let’s get our hands dirty and expand the expression step-by-step. We'll take each term in the first parenthesis and multiply it by every term in the second parenthesis. Here’s how we’ll break it down:

1. Multiply $x^7$ by each term in the second parenthesis:

   • $x^7 * x^7 = x^{7+7} = x^{14}$
   • $x^7 * x^9 = x^{7+9} = x^{16}$
   • $x^7 * x^{12} = x^{7+12} = x^{19}$

2. Multiply $x^9$ by each term in the second parenthesis:

   • $x^9 * x^7 = x^{9+7} = x^{16}$
   • $x^9 * x^9 = x^{9+9} = x^{18}$
   • $x^9 * x^{12} = x^{9+12} = x^{21}$

3. Multiply $x^{12}$ by each term in the second parenthesis:

   • $x^{12} * x^7 = x^{12+7} = x^{19}$
   • $x^{12} * x^9 = x^{12+9} = x^{21}$
   • $x^{12} * x^{12} = x^{12+12} = x^{24}$

Now, let’s write out all the terms we’ve calculated:

$x^{14} + x^{16} + x^{19} + x^{16} + x^{18} + x^{21} + x^{19} + x^{21} + x^{24}$

Great job! We’ve completed the expansion. Next up, we’ll simplify this expression by combining like terms.

Combining Like Terms

The next step in simplifying our expression is to identify and combine like terms. Like terms are those that have the same variable raised to the same power. In our expanded expression:

$x^{14} + x^{16} + x^{19} + x^{16} + x^{18} + x^{21} + x^{19} + x^{21} + x^{24}$

We can see that some terms appear more than once. Let’s group them together:

• $x^{14}$: Appears once
• $x^{16}$: Appears twice
• $x^{19}$: Appears twice
• $x^{18}$: Appears once
• $x^{21}$: Appears twice
• $x^{24}$: Appears once

Now, we combine these like terms by adding their coefficients. Since each term has a coefficient of 1, we simply add the terms together:

• $x^{14}$ remains as $x^{14}$
• $x^{16} + x^{16} = 2x^{16}$
• $x^{19} + x^{19} = 2x^{19}$
• $x^{18}$ remains as $x^{18}$
• $x^{21} + x^{21} = 2x^{21}$
• $x^{24}$ remains as $x^{24}$

So, after combining like terms, our simplified expression looks like this:

$x^{14} + 2x^{16} + 2x^{19} + x^{18} + 2x^{21} + x^{24}$

We usually write polynomials in descending order of exponents, so let’s rearrange the terms:

$x^{24} + 2x^{21} + 2x^{19} + x^{18} + 2x^{16} + x^{14}$

Voila! We’ve simplified the expression. This is the final form of the expansion.

The Final Simplified Expression

After going through all the steps, the simplified form of the expression $(x^7 + x^9 + x^{12})^2$ is:

$x^{24} + 2x^{21} + 2x^{19} + x^{18} + 2x^{16} + x^{14}$

It might seem like a long journey, but breaking it down into smaller steps made it manageable. Remember, practice makes perfect! The more you work with these types of expressions, the easier they become.

Tips for Handling Similar Problems

Now that you’ve conquered this expression, let’s talk about some tips and tricks that can help you with similar problems in the future. Here are a few things to keep in mind:

1. Stay Organized: As you saw, expanding expressions with multiple terms can get messy. Keeping your work organized is crucial. Write each step clearly and double-check your calculations.
2. Master the Distributive Property: The distributive property is the backbone of expanding expressions. Make sure you understand how to apply it correctly.
3. Know Your Exponent Rules: Exponent rules are your best friends when dealing with algebraic expressions. Remember that $x^a * x^b = x^{a+b}$, and use it wisely.
4. Combine Like Terms Carefully: Make sure you’re only combining terms with the same variable and exponent. Double-check to avoid mistakes.
5. Practice, Practice, Practice: The more you practice, the more comfortable you’ll become with these types of problems. Try different expressions and challenge yourself.

Common Mistakes to Avoid

To help you on your algebraic journey, let’s also discuss some common mistakes that students often make when dealing with expressions like this. Avoiding these pitfalls will save you a lot of headaches:

• Forgetting to Distribute: One of the most common mistakes is not distributing every term correctly. Remember, each term in the first parenthesis needs to be multiplied by each term in the second parenthesis.
• Incorrectly Applying Exponent Rules: A wrong application of exponent rules can lead to incorrect answers. Make sure you remember that you add exponents when multiplying terms with the same base, and be careful not to multiply the exponents.
• Missing Like Terms: Sometimes, it’s easy to overlook like terms, especially in longer expressions. Take your time and systematically identify and combine them.
• Simple Arithmetic Errors: It’s easy to make small arithmetic mistakes when dealing with multiple terms. Double-check your calculations to avoid these errors.

Real-World Applications

You might be wondering,