Simplifying (21x² + 9x) ÷ (-3x): A Step-by-Step Guide

Hey guys! Today, we're diving into a common algebra problem: simplifying the expression (21x² + 9x) ÷ (-3x). This might look intimidating at first, but don't worry, we'll break it down step-by-step so you can tackle it with confidence. This type of problem is fundamental in algebra and understanding how to simplify expressions like these will help you in more advanced math topics. So, let’s get started and make sure you understand every single step.

Understanding the Basics: Dividing Polynomials

Before we jump into the specifics of our problem, let's quickly review the basics of dividing polynomials. Think of polynomials as expressions with variables and coefficients, like our (21x² + 9x). Dividing polynomials involves distributing the divisor (in our case, (-3x))) to each term in the dividend. It's super important to understand the rules of exponents here, especially when dividing terms with variables. Remember, when you divide terms with the same base, you subtract the exponents. For example, x² / x = x^(2-1) = x. Also, remember your basic arithmetic rules for dividing positive and negative numbers. A positive divided by a negative is a negative, and a negative divided by a negative is a positive. Keeping these basics in mind will make the process much smoother. This foundation is crucial for more complex algebraic manipulations, so ensuring you're solid on these concepts will pay off in the long run.

Step 1: Rewriting the Expression

The first thing we want to do is rewrite the division problem as a fraction. This makes it visually easier to see what we're working with. So, (21x² + 9x) ÷ (-3x) becomes _ (21x² + 9x) / (-3x)_. This simple change in notation can often make the problem seem less daunting. By expressing the division as a fraction, we set the stage for simplifying each term individually. This is a standard technique in algebra and is applicable to various types of division problems. Seeing the problem in this format helps to clarify the next steps involved in the simplification process. It’s all about making the complex look simple, right?

Step 2: Distributing the Division

Now comes the crucial step: distributing the division. We're going to divide each term in the numerator (the top part of the fraction) by the denominator (the bottom part). This means we'll split the fraction into two separate fractions: (21x² / -3x) + (9x / -3x). By separating the terms, we make the division process more manageable. This technique is a direct application of the distributive property of division over addition. Each term in the numerator is now clearly paired with the denominator, which sets us up perfectly for the next step – simplification. Remember, it's all about breaking down the problem into smaller, more digestible chunks.

Step 3: Simplifying Each Term

This is where the magic happens! Let's simplify each fraction individually.

• First Term: 21x² / -3x. Divide the coefficients: 21 divided by -3 is -7. Now, let's tackle the variables. We have x² divided by x. Remember the rule: when dividing exponents with the same base, you subtract them. So, x² / x = x^(2-1) = x. Combining these, we get -7x. This part highlights the importance of knowing your exponent rules. Without them, simplifying algebraic expressions becomes much harder. So, make sure you're comfortable with these rules!
• Second Term: 9x / -3x. Again, divide the coefficients: 9 divided by -3 is -3. Now, for the variables, we have x divided by x, which is simply 1 (since any non-zero number divided by itself is 1). So, this term simplifies to -3. Notice how the variable completely disappears in this case. This is a common occurrence in simplification problems, and it's important to recognize when variables cancel out.

Step 4: Combining the Simplified Terms

We're almost there! Now we just need to combine the simplified terms we found in the previous step. We have -7x and -3. So, the final simplified expression is -7x - 3. And that's it! We've successfully simplified the original expression. This step demonstrates how simplification often involves bringing together the results of smaller operations. It's like assembling the pieces of a puzzle to see the complete picture. The final result is a cleaner, more concise form of the original expression, making it easier to work with in further calculations or problem-solving.

Common Mistakes to Avoid

Simplifying expressions can be tricky, so let's talk about some common pitfalls to avoid:

• Forgetting to Distribute: One of the biggest mistakes is not distributing the division to every term in the numerator. Remember, each term needs to be divided by the denominator.
• Incorrectly Applying Exponent Rules: Make sure you're subtracting exponents correctly when dividing variables with the same base. A wrong exponent can throw off the entire answer.
• Sign Errors: Pay close attention to the signs (positive and negative) when dividing. A simple sign error can lead to an incorrect result.
• Skipping Steps: It's tempting to rush through the problem, but skipping steps can lead to mistakes. Take your time and write out each step clearly.

By being aware of these common mistakes, you can significantly reduce your chances of making errors. Practice is key, and the more you work through these types of problems, the better you'll become at spotting potential pitfalls.

Practice Problems

Want to put your skills to the test? Here are a few practice problems you can try:

1. (15a² + 6a) ÷ (3a)
2. (24b³ - 18b²) ÷ (-6b)
3. (35x²y + 14xy²) ÷ (7xy)

Work through these problems using the steps we've discussed. Remember to show your work, and don't be afraid to double-check your answers. The more you practice, the more confident you'll become in your ability to simplify expressions.

Conclusion

So, there you have it! Simplifying expressions like (21x² + 9x) ÷ (-3x) might seem daunting initially, but by breaking it down into manageable steps, it becomes much easier. Remember to rewrite the expression as a fraction, distribute the division, simplify each term individually, and then combine the simplified terms. Keep an eye out for those common mistakes, and don't forget to practice! With a little effort, you'll be simplifying algebraic expressions like a pro. You got this, guys! Understanding these fundamental algebraic principles is crucial for success in more advanced math courses. So, keep practicing, keep learning, and keep pushing your mathematical boundaries! And remember, math can be fun when you approach it step by step. Good luck with your algebraic adventures!