Multiplying Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of algebra to tackle the expression $21x(x-7)$. Multiplying expressions might seem intimidating at first, but trust me, with a little practice and understanding of the distributive property, you'll be acing these problems in no time. We'll break down the steps, explain the reasoning behind each one, and provide some helpful tips along the way. Get ready to flex those math muscles!

Understanding the Basics: The Distributive Property

Before we jump into the specific problem, let's quickly review the distributive property. This is the cornerstone of multiplying expressions like the one we're dealing with. Simply put, the distributive property states that you multiply a term outside the parentheses by each term inside the parentheses. Mathematically, it looks like this: a(b + c) = ab + ac.

So, in our expression, $21x(x-7)$, we have 21x outside the parentheses and (x - 7) inside. This means we'll need to multiply 21x by both x and -7. It's like spreading the love of multiplication to everyone inside the parentheses. Don't forget that when multiplying variables, if they have the same base, you add their exponents. For example, x * x = x^2, since x is the same as x^1. Remembering these basic rules is really essential to get started. Don't worry if it's new to you, we'll go through it slowly and step by step. I am pretty sure that after reading this article, you will be able to solve similar problems on your own, guys!

Step-by-Step Solution of $21x(x-7)$

Now, let's get our hands dirty and solve this expression step by step. We'll break it down so you can follow along easily. Remember, the goal is to eliminate the parentheses by applying the distributive property. Let's start!

Step 1: Multiply 21x by x

First, we multiply 21x by x. This gives us 21x * x. When you multiply variables with the same base, you add their exponents. Since both x terms have an exponent of 1 (remember, if there's no exponent written, it's assumed to be 1), this becomes 21x². So, our expression starts to look like this: 21x² .... Pretty straightforward, right?

Keep in mind that when multiplying a number with a variable that has a coefficient, you multiply the coefficients together and keep the variable the same. In this case, 21 times the imaginary coefficient of 1 from the x. Therefore, you have 21 x^2.

Step 2: Multiply 21x by -7

Next, we multiply 21x by -7. Multiplying a positive number by a negative number results in a negative number. So, 21x * -7 equals -147x. Now our expression is: 21x² - 147x. Notice that we are multiplying the constants, which are the regular numbers, and then adding the variable to it. It's really easy when you know the rules!

Step 3: Combine the Results

We've completed the multiplication steps. Now, we simply combine the results from steps 1 and 2. This gives us the final answer: 21x² - 147x. We cannot simplify this further because the terms are not like terms (one term has x² and the other has x). So, this is our simplified expression. Congratulations, we've successfully multiplied the expression! Seems easy now, huh?

Key Takeaways and Tips for Multiplying Expressions

Let's recap what we've learned and provide some helpful tips to make your journey into algebra smoother. Here are some key points to remember when dealing with the multiplication of expressions.

• The Distributive Property is Key: Always remember to distribute the term outside the parentheses to each term inside. This is the foundation of the process.
• Pay Attention to Signs: Be extra careful with positive and negative signs. A small mistake here can change your entire answer. Remember that multiplying a positive number by a negative number gives a negative result, and multiplying two negative numbers gives a positive result. Keep practicing, and you will get used to it.
• Combine Like Terms: After multiplying, always check if you can combine any like terms. Like terms have the same variable raised to the same power. In our example, we couldn't combine them, but in other problems, you might have terms like 3x and 5x which can be combined to form 8x.
• Double-Check Your Work: Always review your steps to avoid any calculation errors. It's easy to miss a negative sign or make a small multiplication mistake, so take your time and be thorough. When solving the problems, make sure that you do the process step by step, so that it becomes easy to check it.

Common Mistakes to Avoid

Even seasoned math lovers can stumble upon common pitfalls. Let's highlight some mistakes that often pop up when multiplying expressions, so you know what to watch out for.

• Forgetting to Distribute: The most common mistake is forgetting to distribute the term outside the parentheses to every term inside. Make sure you multiply by each term to avoid missing out on parts of the expression.
• Incorrect Sign Handling: Mismanaging positive and negative signs is another frequent error. Always double-check your signs, especially when multiplying by negative numbers.
• Incorrect Exponent Rules: Sometimes, students struggle with the rules of exponents. Remember that when multiplying variables with the same base, you add their exponents. For example, x * x = x².
• Combining Unlike Terms: Never try to combine unlike terms (terms that don't have the same variable and exponent). For example, x² and x cannot be combined. They're different types of terms and must be kept separate. Make sure you understand the basics before solving the problems.

Practice Makes Perfect: More Examples

Ready to get some more practice? Let's work through a few more examples to cement your understanding. Remember, the more you practice, the easier and more comfortable you'll become with multiplying expressions. The main idea is always the same: apply the distributive property, pay close attention to signs, and combine like terms if possible.

Example 1: Multiplying with Negative Numbers

Let's try multiplying (-3)(x + 4). Following the distributive property:

• Multiply (-3) by x: (-3) * x = -3x
• Multiply (-3) by 4: (-3) * 4 = -12

Combine the results: -3x - 12. Notice how important it is to keep track of the signs. It's really easy to get confused.

Example 2: More Complex Expressions

How about (2x)(3x - 5)? Here's how to solve it:

• Multiply 2x by 3x: 2x * 3x = 6x²
• Multiply 2x by -5: 2x * -5 = -10x

Combine the results: 6x² - 10x. See how each problem builds on the previous one? It's like putting together pieces of a puzzle. The more you work on it, the better you will get!

Conclusion: Mastering the Art of Multiplication

And there you have it, guys! We've successfully navigated the multiplication of expressions, learned the ins and outs of the distributive property, and hopefully, demystified the process for you. Remember, math is like a muscle – the more you exercise it, the stronger it gets. Keep practicing, don't be afraid to ask for help, and you'll be multiplying expressions like a pro in no time.

Whether you're preparing for a test, working on homework, or simply trying to brush up on your algebra skills, remember that consistency and understanding are your best friends. Keep these tips and examples in mind, and you'll be well on your way to mathematical success. Happy multiplying!

I hope that this article was helpful for you. If you have any more questions, feel free to ask! Good luck and have fun solving the problems!