Solving Polynomial Equation: A Step-by-Step Guide
Hey guys! Let's dive into solving this polynomial equation. It looks a bit intimidating at first, but we'll break it down step by step to make it super easy to understand. Our main goal is to simplify the equation: (x²+2x+5)(x²-3x-4) - (x²-2x-1)(x²+3x+1) - (x²+x+1)(x²-x-1). We’ll walk through each part, explaining the hows and whys, so you can tackle similar problems with confidence. Remember, math can be fun when you break it down into manageable chunks! So, grab your pencils, and let's get started!
Expanding the First Term: (x²+2x+5)(x²-3x-4)
Okay, let's kick things off by tackling the first term: (x²+2x+5)(x²-3x-4). To expand this, we'll use the distributive property, which basically means each term in the first parenthesis multiplies every term in the second parenthesis. This might sound a little tedious, but trust me, it's just a methodical process. We're going to take it nice and slow to make sure we get every detail right.
First, we multiply x² from the first parenthesis with each term in the second parenthesis:
- x² * x² = x⁴
- x² * -3x = -3x³
- x² * -4 = -4x²
Next, we multiply 2x from the first parenthesis with each term in the second parenthesis:
- 2x * x² = 2x³
- 2x * -3x = -6x²
- 2x * -4 = -8x
Finally, we multiply 5 from the first parenthesis with each term in the second parenthesis:
- 5 * x² = 5x²
- 5 * -3x = -15x
- 5 * -4 = -20
Now, let's put it all together: x⁴ - 3x³ - 4x² + 2x³ - 6x² - 8x + 5x² - 15x - 20. This might look like a jumbled mess, but the next step is to combine like terms, which will help us simplify things considerably. Remember, staying organized is key in math, so keep those terms aligned!
Combining Like Terms in the First Expansion
Now that we've expanded (x²+2x+5)(x²-3x-4), we have this long expression: x⁴ - 3x³ - 4x² + 2x³ - 6x² - 8x + 5x² - 15x - 20. Time to make it look prettier by combining those like terms! Like terms are terms that have the same variable raised to the same power. Think of it like sorting your socks – you put all the same types together. In this case, we'll group the x⁴ terms, the x³ terms, the x² terms, the x terms, and the constants.
Let's start with the highest power, x⁴. We only have one x⁴ term, so that's easy: x⁴.
Next up are the x³ terms. We have -3x³ and +2x³. Combining them gives us -3x³ + 2x³ = -x³.
Now for the x² terms: -4x², -6x², and +5x². Combining them, we get -4x² - 6x² + 5x² = -5x².
Moving on to the x terms: -8x and -15x. Adding those together, we have -8x - 15x = -23x.
Finally, the constant term: we only have -20, so that stays as is.
Putting it all together, the simplified expression for the first term is: x⁴ - x³ - 5x² - 23x - 20. See? Much cleaner already! Remember, taking the time to combine like terms not only simplifies the expression but also makes the next steps much easier. It's like decluttering your workspace before starting a big project.
Expanding the Second Term: (x²-2x-1)(x²+3x+1)
Alright, let's move on to the second term: (x²-2x-1)(x²+3x+1). We’re going to use the same distributive property method we used earlier, ensuring that every term in the first parenthesis multiplies every term in the second parenthesis. Think of it as systematically pairing each item from one set with each item from another set. This ensures we don't miss anything and get an accurate expansion.
First, we multiply x² from the first parenthesis with each term in the second parenthesis:
- x² * x² = x⁴
- x² * 3x = 3x³
- x² * 1 = x²
Next, we multiply -2x from the first parenthesis with each term in the second parenthesis:
- -2x * x² = -2x³
- -2x * 3x = -6x²
- -2x * 1 = -2x
Finally, we multiply -1 from the first parenthesis with each term in the second parenthesis:
- -1 * x² = -x²
- -1 * 3x = -3x
- -1 * 1 = -1
Now, let’s put all the terms together: x⁴ + 3x³ + x² - 2x³ - 6x² - 2x - x² - 3x - 1. Just like before, this looks a bit cluttered, but we're about to tidy it up by combining like terms. Keeping things organized at this stage is super helpful for avoiding mistakes later on. So, let's get to sorting those terms!
Combining Like Terms in the Second Expansion
After expanding (x²-2x-1)(x²+3x+1), we've landed at this expression: x⁴ + 3x³ + x² - 2x³ - 6x² - 2x - x² - 3x - 1. Now, just like we did with the first term, we need to combine the like terms to simplify things. Remember, like terms have the same variable raised to the same power, so we'll group them together methodically.
Starting with the x⁴ terms, we see only one: x⁴.
Next, let’s tackle the x³ terms: 3x³ and -2x³. Combining these, we get 3x³ - 2x³ = x³.
Now for the x² terms: x², -6x², and -x². Adding these together, we have x² - 6x² - x² = -6x².
Moving on to the x terms: -2x and -3x. Combining them, we get -2x - 3x = -5x.
Finally, the constant term: we have only -1, so it remains as -1.
Putting everything together, the simplified expression for the second term is: x⁴ + x³ - 6x² - 5x - 1. Awesome! We've trimmed it down nicely. Remember, taking the time to combine like terms is like putting all your ingredients in place before you start cooking – it makes the whole process smoother and less error-prone.
Expanding the Third Term: (x²+x+1)(x²-x-1)
Time to tackle the third term in our equation: (x²+x+1)(x²-x-1). We're sticking with the distributive property method, which by now should be feeling pretty familiar! Remember, this means each term in the first parenthesis needs to multiply every term in the second parenthesis. It’s all about being systematic and making sure we cover every possible pairing.
Let’s start by multiplying x² from the first parenthesis with each term in the second parenthesis:
- x² * x² = x⁴
- x² * -x = -x³
- x² * -1 = -x²
Next, we multiply x from the first parenthesis with each term in the second parenthesis:
- x * x² = x³
- x * -x = -x²
- x * -1 = -x
Finally, we multiply 1 from the first parenthesis with each term in the second parenthesis:
- 1 * x² = x²
- 1 * -x = -x
- 1 * -1 = -1
Now, let’s bring all these terms together: x⁴ - x³ - x² + x³ - x² - x + x² - x - 1. It looks a bit long, but we know what to do – combine those like terms to make it simpler! Just like tidying up after a big cooking session, combining like terms helps us see the result more clearly and prevents any confusion.
Combining Like Terms in the Third Expansion
After expanding (x²+x+1)(x²-x-1), we're here with: x⁴ - x³ - x² + x³ - x² - x + x² - x - 1. Time to roll up our sleeves and combine those like terms! As you know, we're looking for terms with the same variable raised to the same power. Grouping them together is the key to simplifying this expression.
Let's start with the x⁴ terms. We've got only one: x⁴.
Now, let’s move to the x³ terms: -x³ and +x³. When we combine them, we get -x³ + x³ = 0. So, the x³ terms cancel out!
Next are the x² terms: -x², -x², and +x². Combining these, we have -x² - x² + x² = -x².
Now for the x terms: -x and -x. Adding these gives us -x - x = -2x.
Finally, the constant term: we only have -1, so it remains -1.
Putting it all together, the simplified expression for the third term is: x⁴ - x² - 2x - 1. Nice and tidy! By now, you've probably noticed how crucial this step is. Combining like terms not only simplifies the expression but also makes the final calculations much easier.
Putting It All Together and Simplifying
Okay, guys, we've expanded and simplified each term individually. Now comes the exciting part: putting it all together and seeing what we get! Remember our original equation? It was:
(x²+2x+5)(x²-3x-4) - (x²-2x-1)(x²+3x+1) - (x²+x+1)(x²-x-1)
We've simplified each part to:
- (x²+2x+5)(x²-3x-4) = x⁴ - x³ - 5x² - 23x - 20
- (x²-2x-1)(x²+3x+1) = x⁴ + x³ - 6x² - 5x - 1
- (x²+x+1)(x²-x-1) = x⁴ - x² - 2x - 1
Now, let’s plug these back into the original equation. Be super careful with the signs, especially because we're subtracting some of these expressions. It’s like balancing a chemical equation – one wrong sign can change everything!
So, we have:
(x⁴ - x³ - 5x² - 23x - 20) - (x⁴ + x³ - 6x² - 5x - 1) - (x⁴ - x² - 2x - 1)
First, let's distribute the negative signs:
x⁴ - x³ - 5x² - 23x - 20 - x⁴ - x³ + 6x² + 5x + 1 - x⁴ + x² + 2x + 1
Now, it’s time to combine like terms again. We'll group the x⁴ terms, x³ terms, x² terms, x terms, and constants.
Final Simplification: Combining All Terms
Alright, we’ve got everything laid out: x⁴ - x³ - 5x² - 23x - 20 - x⁴ - x³ + 6x² + 5x + 1 - x⁴ + x² + 2x + 1. It’s a bit of a marathon, but we’re in the home stretch! Now, let’s combine those like terms one last time to get to our final, simplified expression.
First, the x⁴ terms: x⁴ - x⁴ - x⁴. This simplifies to x⁴ - x⁴ - x⁴ = -x⁴.
Next, the x³ terms: -x³ - x³. This gives us -x³ - x³ = -2x³.
Now for the x² terms: -5x² + 6x² + x². Combining these, we have -5x² + 6x² + x² = 2x².
Moving on to the x terms: -23x + 5x + 2x. Adding these up, we get -23x + 5x + 2x = -16x.
Finally, the constants: -20 + 1 + 1. This simplifies to -20 + 1 + 1 = -18.
So, our fully simplified expression is: -x⁴ - 2x³ + 2x² - 16x - 18. Wow, we made it! That’s a lot cleaner than where we started, right? This process might seem long, but each step is manageable, and the final result is well worth the effort. Remember, tackling complex problems is all about breaking them down into smaller, easier steps. Great job sticking with it!
Conclusion: The Final Simplified Polynomial
And there you have it, guys! After a thorough step-by-step journey through expanding and simplifying, we've arrived at our final answer. The simplified form of the polynomial equation (x²+2x+5)(x²-3x-4) - (x²-2x-1)(x²+3x+1) - (x²+x+1)(x²-x-1) is -x⁴ - 2x³ + 2x² - 16x - 18.
We started with a complex-looking equation, but by breaking it down into smaller, manageable parts, we were able to tackle each section methodically. We expanded each term, combined like terms, and finally put everything together to get our simplified result. Remember, the key to solving these types of problems is organization, patience, and a systematic approach. Keep practicing, and you'll become a pro at simplifying polynomials in no time!
If you found this guide helpful, keep practicing similar problems, and don't hesitate to review the steps whenever you need a refresher. Happy solving!