Coefficient Of X In (3x+4)(5x^2+x-2)? Explained!

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Hey guys! Let's dive into this math problem where we need to figure out the coefficient of the x term after expanding the expression (3x+4)(5x^2+x-2). It might look a bit intimidating at first, but trust me, we'll break it down step-by-step so it's super easy to understand. So, grab your pencils and let's get started!

Expanding the Expression

Okay, so the first thing we need to do is expand the expression. This means we're going to multiply each term in the first set of parentheses by each term in the second set. Think of it like a math party where everyone gets to mingle! We'll use the distributive property, which is just a fancy way of saying we're going to multiply each term correctly. Remember PEMDAS! In mathematics, PEMDAS is a mnemonic order of operations, where letters represent Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction. Proper application of the order of operations is very important to obtain the correct result.

Let's start by multiplying 3x by each term in the second parentheses:

  • 3x * 5x^2 = 15x^3
  • 3x * x = 3x^2
  • 3x * -2 = -6x

Now, let's multiply 4 by each term in the second parentheses:

  • 4 * 5x^2 = 20x^2
  • 4 * x = 4x
  • 4 * -2 = -8

Great! Now we have all the individual products. Let's put them all together:

15x^3 + 3x^2 - 6x + 20x^2 + 4x - 8

Combining Like Terms

The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. Think of them as the same family of terms. We can only add or subtract terms that are in the same family. For instance, the x^2 terms can be combined, and the x terms can be combined, but we can't combine an x^2 term with an x term.

Let's identify and combine our like terms:

  • x^3 terms: We only have one x^3 term, which is 15x^3. So, it stays as is.
  • x^2 terms: We have 3x^2 and 20x^2. Combining them gives us 3x^2 + 20x^2 = 23x^2.
  • x terms: We have -6x and 4x. Combining them gives us -6x + 4x = -2x.
  • Constant terms: We only have one constant term, which is -8. So, it stays as is.

Putting it all together, our simplified expression is:

15x^3 + 23x^2 - 2x - 8

Identifying the Coefficient of the x Term

Alright, we're in the home stretch! The question asks for the coefficient of the x term. Remember, the coefficient is the number that's multiplied by the variable. In our simplified expression, the x term is -2x. So, the coefficient of the x term is -2. Easy peasy!

Therefore, the answer is C. -2

Why is This Important?

You might be thinking, “Okay, I can do this problem, but why do I even need to know this?” Well, understanding how to expand expressions and identify coefficients is crucial in many areas of math and science. It's a fundamental skill that you'll use in algebra, calculus, physics, and even computer science! Knowing how to manipulate expressions like this helps you solve equations, model real-world situations, and understand complex concepts. Think of it as building a solid foundation for your future studies. This skillset will help you to understand polynomial functions and how the coefficients affect the shape and behavior of the graph. Plus, it hones your algebraic manipulation skills, which are essential for tackling more advanced mathematical challenges.

Common Mistakes to Avoid

Let's chat about some common pitfalls folks often encounter when tackling problems like this. Being aware of these can save you from making simple errors and boost your confidence!

  • Forgetting to Distribute: One of the biggest slip-ups is not multiplying every term inside the parentheses. Remember, each term in the first set of parentheses needs to be multiplied by each term in the second set. Imagine skipping someone at a party – we don't want that! Double-check that you've distributed correctly.
  • Combining Unlike Terms: This is a classic! You can only combine terms that have the same variable raised to the same power. It's like trying to add apples and oranges – they're just not the same. Make sure you're only adding or subtracting like terms.
  • Sign Errors: Watch out for those pesky negative signs! They can be tricky. A negative multiplied by a negative is positive, and a negative multiplied by a positive is negative. Keep a close eye on the signs as you're working through the problem.
  • Order of Operations: Remember PEMDAS! Make sure you're following the correct order of operations. Multiplication comes before addition and subtraction. Getting the order wrong can lead to a completely different answer.
  • Rushing: It’s tempting to rush through a problem, especially if you feel confident. But taking your time and working carefully can prevent silly mistakes. Double-check each step as you go.

By being mindful of these common errors, you can improve your accuracy and problem-solving skills. Remember, practice makes perfect! The more you work through these types of problems, the more comfortable and confident you'll become.

Practice Problems

Want to really nail this skill? Here are a few practice problems you can try. Working through these will help solidify your understanding and build your confidence.

  1. Simplify: (2x - 1)(x^2 + 3x - 4)
  2. What is the coefficient of the x term in the simplified expression of (x + 2)(4x^2 - x + 3)?
  3. Expand and simplify: (3x - 2)(2x^2 - 5x - 1)

Try working through these on your own. Don't be afraid to make mistakes – that's how we learn! If you get stuck, go back and review the steps we discussed earlier. And remember, there are tons of resources online if you need extra help.

Conclusion

So, there you have it! We've successfully expanded the expression, combined like terms, and identified the coefficient of the x term. You're now equipped to tackle similar problems with confidence. Remember, practice is key, so keep working at it, and you'll become a pro in no time! Keep up the awesome work, and I'll catch you in the next math adventure!