Product Of Polynomials: Solving (x^4)(3x^3-2)(4x^2+5x)

by Admin 55 views
Product of Polynomials: Solving (x^4)(3x^3-2)(4x^2+5x)

Hey guys! Let's dive into a fun math problem today. We're going to figure out the product of the polynomial expression: (x4)(3x3-2)(4x^2+5x). Don't worry, it looks scarier than it is! We'll break it down step-by-step so it's super easy to follow.

Understanding Polynomial Multiplication

Before we jump into the actual problem, let's quickly recap what polynomial multiplication is all about. Polynomials are expressions with variables and coefficients, like our 3x^3 - 2 or 4x^2 + 5x. When we multiply polynomials, we're essentially applying the distributive property multiple times. This means each term in one polynomial needs to be multiplied by each term in the other polynomial. Think of it as making sure everyone gets a handshake!

The key idea here is to multiply the coefficients (the numbers in front of the variables) and add the exponents (the little numbers above the variables). Remember the rule: x^m * x^n = x^(m+n). This will be crucial as we work through our problem. Also, it is very important to be careful with the signs. A negative multiplied by a positive results in a negative, and so on.

Step-by-Step Solution

Okay, let's tackle our problem: (x4)(3x3-2)(4x^2+5x). To solve this, we’ll multiply the polynomials in a specific order to keep things organized and avoid confusion. It's like following a recipe – each step leads to the final delicious result!

Step 1: Multiply the first two factors

First, we'll multiply the first two factors: (x^4) and (3x^3 - 2). This involves distributing x^4 across both terms inside the parenthesis:

  • x^4 * 3x^3 = 3x^(4+3) = 3x^7
  • x^4 * -2 = -2x^4

So, (x^4)(3x^3 - 2) = 3x^7 - 2x^4. Now we have a new, simplified expression to work with. See? We're making progress!

Step 2: Multiply the result by the third factor

Next, we'll take our result, (3x^7 - 2x^4), and multiply it by the third factor, (4x^2 + 5x). This is where the distributive property really shines. We need to multiply each term in the first expression by each term in the second expression:

  • Multiply 3x^7 by both terms in (4x^2 + 5x):
    • 3x^7 * 4x^2 = 12x^(7+2) = 12x^9
    • 3x^7 * 5x = 15x^(7+1) = 15x^8
  • Multiply -2x^4 by both terms in (4x^2 + 5x):
    • -2x^4 * 4x^2 = -8x^(4+2) = -8x^6
    • -2x^4 * 5x = -10x^(4+1) = -10x^5

Step 3: Combine the results

Now, let's put all the pieces together. We have:

12x^9 + 15x^8 - 8x^6 - 10x^5

This is the final expanded form of our polynomial expression. We've successfully multiplied everything out!

Analyzing the Options

Now that we have our answer, let's look at the options provided and see which one matches:

A. 12x^9 + 15x^8 - 8x^6 - 10x^5 B. 12x^24 + 15x^12 - 8x^8 - 10x^4 C. 12x^9 - 10x^5 D. 12x^24 - 10x^4

As you can clearly see, option A, 12x^9 + 15x^8 - 8x^6 - 10x^5, is the correct answer. We did it!

Common Mistakes to Avoid

Polynomial multiplication is pretty straightforward once you get the hang of it, but there are a few common pitfalls to watch out for:

  • Forgetting to Distribute: Make sure you multiply every term in one polynomial by every term in the other. It's easy to miss one, especially when there are multiple terms.
  • Incorrectly Adding Exponents: Remember, when multiplying terms with the same base (like x), you add the exponents, not multiply them. For example, x^2 * x^3 = x^(2+3) = x^5, not x^6.
  • Sign Errors: Keep a close eye on your signs (positive and negative). A single sign error can throw off your entire answer.
  • Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For instance, 3x^2 and 5x^2 can be combined, but 3x^2 and 5x^3 cannot.

Why This Matters: Real-World Applications

You might be thinking, “Okay, this is cool, but when am I ever going to use this in real life?” Well, polynomial multiplication isn't just an abstract math concept. It has practical applications in various fields, including:

  • Engineering: Engineers use polynomials to model curves and surfaces, calculate areas and volumes, and analyze systems.
  • Computer Graphics: Polynomials are essential for creating smooth curves and surfaces in computer graphics and animations.
  • Economics: Economists use polynomial functions to model cost, revenue, and profit.
  • Physics: Polynomials appear in many physics equations, such as those describing projectile motion.

So, while you might not be solving polynomial multiplication problems every day, the underlying concepts are used in many different areas. Understanding these concepts builds a strong foundation for more advanced math and science topics.

Practice Makes Perfect

The best way to master polynomial multiplication is to practice, practice, practice! Here are a few extra problems you can try:

  1. (2x^2 + 1)(x - 3)
  2. (x^3 - 2x)(4x + 5)
  3. (x^2 + 3x - 2)(x - 1)

Work through these problems step-by-step, and don't be afraid to double-check your work. If you get stuck, go back and review the steps we covered earlier. With a little bit of effort, you'll become a polynomial multiplication pro!

Conclusion

So, guys, we've successfully navigated the world of polynomial multiplication! We learned how to multiply polynomials step-by-step, identified common mistakes to avoid, and explored some real-world applications. Remember, the key is to take it one step at a time, distribute carefully, and watch those exponents and signs. Keep practicing, and you'll be multiplying polynomials like a champ in no time!

If you have any questions or want to explore more math topics, let me know. Happy calculating!