Expanding Polynomials: A Step-by-Step Guide

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Hey guys! Today, we're diving deep into the world of polynomials and tackling the challenge of expanding a rather complex expression. Specifically, we'll be working through the expansion of (7x2+8)(2x3+5)(x2−4x−9){(7x^2 + 8)(2x^3 + 5)(x^2 - 4x - 9)}. This might seem daunting at first, but trust me, by breaking it down step-by-step, it becomes totally manageable. So, grab your pencils, and let's get started!

Understanding Polynomial Expansion

Before we jump into the nitty-gritty, let's quickly recap what it means to expand polynomials. Essentially, when we expand, we're taking an expression written in a factored form (like the one we have) and multiplying everything out to get a single polynomial expression. This involves using the distributive property repeatedly. Remember that old friend? It's the key to success here! Think of it as carefully distributing each term across the others, ensuring no term is left behind. We need to ensure every term in the first set of parentheses multiplies every term in the second set, and so on. Accuracy is key here, guys. One little slip and the whole thing goes sideways. It's like baking a cake; you can't forget the flour!

Polynomial expansion is a fundamental concept in algebra, and mastering it opens doors to solving more complex equations and problems. You'll encounter it everywhere, from simple quadratic equations to advanced calculus problems. So, investing the time to truly understand this process is an investment in your mathematical future. Plus, it's super satisfying when you finally unravel a big, complicated expression and arrive at the simplified answer. It's like cracking a secret code, guys!

The distributive property, the workhorse of polynomial expansion, states that for any numbers a, b, and c:

a(b + c) = ab + ac

We'll be using this principle over and over again as we tackle our expression. It's the bedrock of our entire process. The challenge here is not just applying the distributive property once, but multiple times, layering it on as we move through the different sets of parentheses. That's where careful organization and a methodical approach become essential. We're not just doing simple multiplication; we're orchestrating a carefully choreographed mathematical dance! And with each step, we get closer to revealing the final, expanded form of our polynomial.

Step 1: Multiplying the First Two Factors

Okay, let's get our hands dirty. Our first task is to multiply the first two factors: (7x2+8)(2x3+5){(7x^2 + 8)(2x^3 + 5)}. We'll use the distributive property (or the FOIL method, which is essentially a specific application of the distributive property for binomials) to do this. FOIL stands for First, Outer, Inner, Last, reminding us to multiply each term in the first binomial by each term in the second.

Here's how it breaks down:

  • First: 7x2∗2x3=14x5{7x^2 * 2x^3 = 14x^5}
  • Outer: 7x2∗5=35x2{7x^2 * 5 = 35x^2}
  • Inner: 8∗2x3=16x3{8 * 2x^3 = 16x^3}
  • Last: 8∗5=40{8 * 5 = 40}

Now, we add these terms together:

14x5+35x2+16x3+40{14x^5 + 35x^2 + 16x^3 + 40}

It's a good practice to write the polynomial in descending order of exponents, so let's rearrange it:

14x5+16x3+35x2+40{14x^5 + 16x^3 + 35x^2 + 40}

Great! We've conquered the first hurdle. This resulting polynomial is our intermediate result, which we'll then use in the next step. Think of it as building a foundation for the rest of our calculation. We've combined the first two pieces of the puzzle, and now we're ready to integrate the final piece. This step-by-step approach is crucial, guys. It keeps things manageable and reduces the chance of making mistakes. Remember, math is like building a skyscraper; a solid foundation is everything!

Step 2: Multiplying the Result by the Third Factor

Now comes the more challenging part. We need to multiply our result from Step 1, 14x5+16x3+35x2+40{14x^5 + 16x^3 + 35x^2 + 40}, by the third factor, (x2−4x−9){(x^2 - 4x - 9)}. This is where things get a little more intricate, but we'll handle it with the same careful approach. We're essentially distributing each term of the first polynomial across all three terms of the second polynomial. It’s a marathon, not a sprint, guys, so let's pace ourselves!

We'll go term by term:

  • 14x5∗(x2−4x−9)=14x7−56x6−126x5{14x^5 * (x^2 - 4x - 9) = 14x^7 - 56x^6 - 126x^5}
  • 16x3∗(x2−4x−9)=16x5−64x4−144x3{16x^3 * (x^2 - 4x - 9) = 16x^5 - 64x^4 - 144x^3}
  • 35x2∗(x2−4x−9)=35x4−140x3−315x2{35x^2 * (x^2 - 4x - 9) = 35x^4 - 140x^3 - 315x^2}
  • 40∗(x2−4x−9)=40x2−160x−360{40 * (x^2 - 4x - 9) = 40x^2 - 160x - 360}

Notice how we're carefully multiplying each term and keeping track of the exponents. That's the key to avoiding errors. It’s like conducting an orchestra, ensuring each instrument plays its part at the right time and in the right way. Now, we have a series of polynomials that we need to combine. This is where the next crucial step comes in: collecting like terms.

Step 3: Combining Like Terms

The next step is crucial for simplifying our expression: combining like terms. This involves identifying terms with the same variable and exponent and then adding their coefficients. It's like sorting a mixed bag of candies, grouping all the chocolates together, all the caramels, and so on. This step brings order to our mathematical chaos and helps us arrive at the final, most simplified form.

Let's gather our terms and line them up neatly:

  • 14x7{14x^7} (No other x7{x^7} term)
  • −56x6{-56x^6} (No other x6{x^6} term)
  • −126x5+16x5=−110x5{-126x^5 + 16x^5 = -110x^5}
  • −64x4+35x4=−29x4{-64x^4 + 35x^4 = -29x^4}
  • −144x3−140x3=−284x3{-144x^3 - 140x^3 = -284x^3}
  • −315x2+40x2=−275x2{-315x^2 + 40x^2 = -275x^2}
  • −160x{-160x} (No other x{x} term)
  • −360{-360} (No other constant term)

We carefully added the coefficients of terms with the same exponent. This is where attention to detail is paramount, guys. A small mistake here can throw off the entire result. Think of it like balancing a checkbook; every single penny counts! Now we have all the pieces, and we're ready to assemble the final answer.

Final Result

After combining like terms, we arrive at the expanded form of our original expression:

14x7−56x6−110x5−29x4−284x3−275x2−160x−360{14x^7 - 56x^6 - 110x^5 - 29x^4 - 284x^3 - 275x^2 - 160x - 360}

And there you have it! We've successfully expanded the polynomial. It might look like a beast, but we tamed it by breaking it down into manageable steps. This is the expanded form of the product (7x2+8)(2x3+5)(x2−4x−9){(7x^2 + 8)(2x^3 + 5)(x^2 - 4x - 9)}. We started with a factored expression, navigated through the complexities of distribution, and emerged with a single, unified polynomial.

Key Takeaway: The key to expanding complex polynomials is to be organized, methodical, and patient. Don't rush the process. Double-check your work at each step. And remember, practice makes perfect! The more you expand polynomials, the more comfortable and confident you'll become. It's like learning a new language; the more you practice, the more fluent you become.

Tips for Success

Here are a few extra tips to keep in mind when expanding polynomials:

  • Stay Organized: Write neatly and keep your terms aligned. This will make it easier to combine like terms later.
  • Double-Check: After each step, take a moment to review your work. It's easier to catch mistakes early on.
  • Take Breaks: If you're feeling overwhelmed, take a break and come back to it with fresh eyes.
  • Practice, Practice, Practice: The more you practice, the better you'll become at expanding polynomials.

Expanding polynomials might seem like a mathematical marathon, but with a steady pace, a clear strategy, and a sprinkle of patience, you can conquer any expression that comes your way. So keep practicing, keep exploring, and keep expanding your mathematical horizons! You got this, guys!