Multiplying Polynomials: A Step-by-Step Guide

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Hey there, math enthusiasts! Today, we're diving into the world of polynomials, specifically how to multiply them. The question we're tackling is: What is the product of (7x2)(2x3 + 5)(x^2 - 4x - 9)? This might seem a bit daunting at first, but trust me, we'll break it down step by step to make it super clear and easy to understand. Let's get started, shall we?

Understanding the Basics: Polynomial Multiplication

Alright, before we jump into the specific problem, let's quickly recap the fundamentals of polynomial multiplication. The core idea is to distribute each term in one polynomial to every term in the other polynomials. Think of it like this: you're essentially ensuring that every component interacts with every other component. When you multiply terms, remember to multiply the coefficients (the numbers in front of the variables) and add the exponents of the variables if they are the same. This is key! This is where most people stumble when they first encounter this concept, so really pay attention to the details. Keep an eye on the signs as well; a negative times a negative equals a positive, and a positive times a negative gives a negative result. It's all about being systematic and organized. This process might seem time-consuming, but with practice, it'll become second nature. Believe it or not, this skill is a building block for more complex math problems, so mastering it now will pay off big time in the long run.

So, the main principle involves the distributive property, which is simply a fancy way of saying that you must multiply each term in the first set of parentheses by each term in the second set, and then multiply the result by each term in the third. It looks complicated, but, it is not, just keep going step by step, and do not try to skip steps. You will get the hang of it pretty quickly.

Let’s start with the given question and see how we solve it: What is the product of (7x2)(2x3 + 5)(x^2 - 4x - 9)? First, we are going to multiply (7x^2) by (2x^3 + 5), and then we will multiply that product with (x^2 - 4x - 9). Does that make sense? Cool, then let's get down to business and start working on the calculations. It’s all about the basics, and the details and keeping them in order. Remember, if we multiply something with another, we will have to use the distributive property.

Step-by-Step Solution: Breaking Down the Problem

Now, let's break down the given expression, (7x^2)(2x^3 + 5)(x^2 - 4x - 9).

Step 1: Multiply the first two polynomials

First, multiply 7x^2 by (2x^3 + 5). We use the distributive property here:

7x^2 * 2x^3 = 14x^5 (Multiply the coefficients and add the exponents: 7 * 2 = 14, and x^2 * x^3 = x^(2+3) = x^5) 7x^2 * 5 = 35x^2 (Multiply the coefficients: 7 * 5 = 35)

So, (7x^2)(2x^3 + 5) = 14x^5 + 35x^2

Step 2: Multiply the result by the third polynomial

Now, multiply the result from Step 1 (14x^5 + 35x^2) by the third polynomial (x^2 - 4x - 9). Again, we use the distributive property. This is where things can get a bit long, so let's be super careful. This step is crucial; don't rush!

14x^5 * x^2 = 14x^7 14x^5 * -4x = -56x^6 14x^5 * -9 = -126x^5 35x^2 * x^2 = 35x^4 35x^2 * -4x = -140x^3 35x^2 * -9 = -315x^2

Step 3: Combine all of the results

Finally, we will combine the results, and, here's what we have now.

Now put it all together.

14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2

The Final Answer

Based on our calculations: The product of (7x^2)(2x^3 + 5)(x^2 - 4x - 9) is 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2. Thus, the correct answer is option C.

Tips for Success: Mastering Polynomial Multiplication

  • Organize Your Work: Write out each step clearly. This helps prevent errors.
  • Pay Attention to Signs: A missed negative sign can throw off the whole answer.
  • Double-Check Your Exponents: Ensure you're adding them correctly.
  • Practice Regularly: The more you practice, the faster and more accurate you'll become.
  • Use the FOIL Method: If you're multiplying two binomials, the FOIL method (First, Outer, Inner, Last) is a handy shortcut. But in this case, we have a trinomial, which is what we did in the above steps.
  • Take Your Time: There's no rush! Work carefully and methodically.

Polynomial multiplication is a fundamental skill in algebra, and with practice, you'll become a pro in no time. Keep practicing and applying these steps, and you’ll find that these kinds of problems become much more manageable.

Common Mistakes to Avoid

  • Forgetting to distribute to every term. This is the most common mistake.
  • Incorrectly adding exponents. Double-check that you're adding them, not multiplying them.
  • Mixing up the signs. A negative times a negative is positive, and so on.
  • Not simplifying completely. Make sure you've combined all like terms and that your final answer is in its simplest form. Remember to double-check that you have multiplied all the terms. Many students forget this step when they are first getting the hang of it, but after a bit of practice, this should be a piece of cake. This whole process might seem a bit long, but, trust me, it’s not that bad.

Conclusion: Keep Practicing

So there you have it, folks! Multiplying polynomials might seem complicated at first, but, with a methodical approach, it's totally doable. Remember to keep practicing, stay organized, and always double-check your work. You've got this! Now, go forth and conquer those polynomial problems! Keep up the great work and the practice, and before you know it, you will be solving them like a boss!