Synthetic Division: Solving (4x⁵+6x⁴+5x²-x-10) ÷ (2x²+3)

Hey math enthusiasts! Let's dive into the world of polynomial division, specifically tackling the problem of dividing (4x⁵ + 6x⁴ + 5x² - x - 10) by (2x² + 3). We're going to use synthetic division, but hold on a sec – there's a slight twist because our divisor isn't in the simple (x - k) format. No worries, though! We'll break it down step by step, making sure you understand every detail. Synthetic division is a cool shortcut for dividing polynomials, but it's usually set up for divisors that are linear (like x - k). When we have a quadratic divisor like 2x² + 3, we need to get a bit creative. Essentially, we'll be using a modified approach to achieve the division. The main idea is to perform a series of operations that will eventually give us our quotient and remainder. It's all about manipulating the coefficients and keeping track of the degrees of the terms. The cool thing about synthetic division is that it streamlines the process, making it less prone to errors compared to the long division method, especially when dealing with higher-degree polynomials. We will get through this together, and by the end of this guide, you'll be able to confidently handle this type of problem. It's all about understanding the process and paying attention to the details. Let's get started! First, we will look into the divisor of the given problem.

Understanding the Problem: The Setup

Alright, before we jump into the nitty-gritty, let's clarify our problem. We're trying to divide the polynomial 4x⁵ + 6x⁴ + 5x² - x - 10 by 2x² + 3. Notice that our divisor (2x² + 3) is a quadratic, not a linear expression in the form (x - k). This means we can't directly apply standard synthetic division. Standard synthetic division is designed for linear divisors. However, we can adapt our approach. The core concept involves finding the quotient and the remainder. When dividing polynomials, the result is always in the form: Dividend = Quotient * Divisor + Remainder. Our goal is to find what the quotient and remainder are in this case. With the given dividend, 4x⁵ + 6x⁴ + 5x² - x - 10, and divisor, 2x² + 3, we will employ a modified approach to achieve this division. The process will involve manipulating the coefficients of the dividend and keeping track of the degrees of the terms, which is a core aspect of synthetic division. Let's set things up so we're ready to solve the problem step-by-step. In polynomial division, we aim to find the quotient and remainder when dividing a dividend by a divisor. When the divisor is not a simple linear expression, we need to use a modified approach. This ensures we handle the problem efficiently and accurately. Let's set up the problem now.

Adapting Synthetic Division: The Modified Approach

Since we're not dealing with a simple (x - k) divisor, we'll use a method that's similar in principle to synthetic division but requires a bit more calculation. We'll focus on dividing the leading terms and systematically working our way through the polynomial. Instead of the typical setup, we will focus on the long division approach to make the calculation easier. This is because direct synthetic division isn't straightforward with a quadratic divisor. We start by dividing the leading term of the dividend (4x⁵) by the leading term of the divisor (2x²), which gives us 2x³. Then, we multiply the entire divisor (2x² + 3) by 2x³ to get 4x⁵ + 6x³. We subtract this result from the original dividend. Don't worry; it sounds more complicated than it is! We're basically mirroring the steps of long division, but keeping things organized. After subtracting, we bring down the next term (-x) to perform the next round of calculation. Synthetic division aims at simplifying the division process. Adapting this method means we have to keep track of the degrees of the terms and manipulate the coefficients accordingly. The key is to break down the division into smaller, manageable steps. Remember, our goal is to find both the quotient and the remainder. The modified approach lets us do exactly that, ensuring we get an accurate result. Let's continue with the next phase of calculations.

Step-by-Step Solution: Unveiling the Quotient and Remainder

Okay, let's break down the division into manageable steps. First, we divide 4x⁵ (from the dividend) by 2x² (from the divisor), which gives us 2x³. This becomes the first term of our quotient. Next, we multiply the entire divisor (2x² + 3) by 2x³, resulting in 4x⁵ + 6x³. Subtract this from the original dividend: (4x⁵ + 6x⁴ + 0x³ + 5x² - x - 10) - (4x⁵ + 0x⁴ + 6x³) = 6x⁴ - 6x³ + 5x² - x - 10. Bring down the next term (-x) as it's necessary for the next stage. Now, we divide the new leading term (6x⁴) by 2x² (from the divisor), which gives us 3x². This is the next term of our quotient. Multiply the divisor (2x² + 3) by 3x², resulting in 6x⁴ + 9x². Subtract this from 6x⁴ - 6x³ + 5x² - x - 10, and you get -6x³ - 4x² - x - 10. Next, divide -6x³ by 2x², which gives us -3x, this is another term in the quotient. Then multiply the divisor (2x² + 3) by -3x, which results in -6x³ - 9x. Subtracting it from the above step leads to -4x² + 8x - 10. Now, divide -4x² by 2x², which results in -2. Multiply the divisor by -2, resulting in -4x² - 6. Subtracting, we get 8x - 4 as the remainder. After each step, we're left with a term that we bring down. With each iteration, we refine our quotient and inch closer to the remainder. Following each step carefully is essential to the calculations, ensuring you don't miss any of the terms. By proceeding methodically, we break down this complex problem into a series of simple operations.

The Final Result: Quotient and Remainder

After completing all the steps, we've arrived at our final answer. The quotient is 2x³ + 3x² - 3x - 2, and the remainder is 8x - 4. The division of (4x⁵ + 6x⁴ + 5x² - x - 10) by (2x² + 3) gives us: 2x³ + 3x² - 3x - 2 + (8x - 4) / (2x² + 3). Remember that the remainder is always expressed over the original divisor. To summarize, the key to this problem was adapting the division to fit the quadratic divisor. Although we couldn’t use a standard synthetic division, we modified our approach to solve the problem. Mastering polynomial division is a valuable skill in algebra, and understanding these steps will greatly help you when you encounter similar problems. The final step is expressing the remainder, so it can be used to check the work. The answer involves the quotient and the remainder, which provides a complete understanding of the division. By now, you should have a clear grasp of how to solve this type of problem. Congratulations on completing this guide; you've learned how to divide a polynomial by a quadratic expression!

Tips for Success: Mastering Polynomial Division

To truly master polynomial division, especially when you encounter problems like the one we just solved, consider these tips. First, practice, practice, practice! The more you work through problems, the more comfortable and efficient you'll become. Try different examples and vary the complexity of the polynomials. Second, always double-check your work. It's easy to make a small arithmetic error, so make sure you are careful with signs and coefficients. Third, when adapting synthetic division or long division, take it step by step and stay organized. This will prevent errors and make it easier to follow your work. Fourth, review the fundamentals. Make sure you're comfortable with exponent rules and basic algebra. Strong foundations will make your journey smoother. Lastly, consider using online resources and tutorials. Videos and interactive tools can provide alternative explanations and visual aids that can help you understand the concepts better. Remember, math is a skill that improves with practice and patience. Keep learning, and don't be afraid to ask for help when you need it. Make sure you understand the steps, from setting up the problem to interpreting the quotient and remainder. Polynomial division is a crucial concept in algebra. By following these tips, you'll be well-equipped to handle various polynomial division problems with confidence and precision. You will become a master of synthetic division. Keep practicing, and you will do great!