Polynomial Division Of (x^3 - 16x^2 - 8x - 13) By (x - 2) A Step-by-Step Guide

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Hey guys! Today, we're diving deep into the world of polynomial division, and trust me, it's not as scary as it sounds. We're going to break down the expression (x^3 - 16x^2 - 8x - 13) á (x - 2). Polynomial division is a fundamental concept in algebra, and mastering it will seriously level up your math game. It's like having a super-power when you're simplifying complex expressions or solving equations. So, grab your calculators, and let's get started!

Understanding Polynomial Division

First things first, what exactly is polynomial division? At its core, it's the process of dividing one polynomial by another. Think of it like long division with numbers, but now we're dealing with variables and exponents. Polynomial division is not only a fundamental skill in algebra but also a stepping stone to more advanced mathematical concepts. It is used extensively in calculus, where understanding the behavior of functions often requires simplifying complex polynomial expressions. Moreover, in various fields of engineering and computer science, polynomial division plays a crucial role in algorithm design and system analysis. For example, in control systems, engineers use polynomial division to analyze the stability of a system by examining the roots of characteristic equations, which are often polynomials. Similarly, in coding theory, polynomial division is used in the error detection and correction codes that ensure reliable data transmission and storage. Therefore, mastering polynomial division opens doors to a multitude of applications beyond the classroom, making it an invaluable tool in your mathematical toolkit. We use it to simplify expressions, solve equations, and even graph functions. There are a couple of ways to tackle this, but the most common methods are long division and synthetic division. Today, we'll focus on long division because it gives you a solid understanding of the process, and it works every time, no matter how complicated the polynomials get.

Setting Up the Problem

Before we jump into the nitty-gritty, let's set up our problem. We have the dividend, which is (x^3 - 16x^2 - 8x - 13), and the divisor, which is (x - 2). Just like with regular long division, we write the dividend inside the division symbol and the divisor outside. Now, make sure your dividend is in descending order of powers of x. In our case, it already is: x^3, x^2, x, and then the constant term. This is super important because it keeps everything organized and prevents mistakes. Also, if you're missing any terms (like if there was no x^2 term), you'd need to add a placeholder with a coefficient of 0. For example, if our dividend was x^3 - 8x - 13, we'd rewrite it as x^3 + 0x^2 - 8x - 13. These placeholders are critical because they maintain the correct spacing for the coefficients and ensure that the division process aligns the like terms properly. Without these placeholders, the division algorithm can easily become misaligned, leading to incorrect results. Furthermore, properly setting up the problem allows for a clear and systematic approach, which is especially important when dealing with higher-degree polynomials or more complex divisors. By taking the time to arrange the terms in descending order and insert placeholders as needed, you are setting the stage for a successful and accurate division process.

Step-by-Step Long Division

Alright, let's get to the fun part – the actual long division! Here’s how it works:

  1. Divide the First Terms: Divide the first term of the dividend (x^3) by the first term of the divisor (x). x^3 á x = x^2. This is the first term of our quotient (the answer). When performing this initial division, you're essentially asking, "What do I need to multiply the first term of the divisor by to get the first term of the dividend?" The answer not only gives you the first term of the quotient but also sets the stage for the subsequent steps. It's crucial to get this first step right, as it influences all the following calculations. A mistake here can propagate through the rest of the division, leading to an incorrect final answer. Moreover, this step highlights the importance of understanding the properties of exponents. When dividing terms with the same base, you subtract the exponents (in this case, x^3 á x^1 = x^(3-1) = x^2). A solid grasp of these properties ensures that you can accurately determine the terms of the quotient.

  2. Multiply: Multiply the entire divisor (x - 2) by the first term of the quotient (x^2). x^2 * (x - 2) = x^3 - 2x^2. This multiplication step is where you distribute the first term of the quotient across the entire divisor. It's like the reverse of factoring—you're expanding a product back into its constituent terms. The result, x^3 - 2x^2, is what you'll use in the next step to subtract from the dividend. This step is also a great opportunity to double-check your work. Make sure you've correctly multiplied each term in the divisor by the quotient term, paying close attention to the signs. A small error in sign can lead to a cascade of mistakes down the line. Moreover, this multiplication helps you see how much of the dividend the first term of the quotient accounts for. It essentially tells you how much of the initial polynomial you've "used up" in this first iteration of the division process.

  3. Subtract: Subtract the result (x^3 - 2x^2) from the corresponding terms in the dividend (x^3 - 16x^2). (x^3 - 16x^2) - (x^3 - 2x^2) = -14x^2. Subtraction is a critical step in long division, as it helps you determine the remainder that needs to be further divided. It's essential to align like terms carefully during this step to avoid mistakes. Subtracting (x^3 - 2x^2) from (x^3 - 16x^2) involves changing the signs of the terms being subtracted and then combining like terms. This process can be tricky, so it's a good idea to take your time and double-check your work. A common mistake is forgetting to distribute the negative sign, which can lead to an incorrect result. This step also sets the stage for bringing down the next term from the dividend, continuing the division process. The result of the subtraction, -14x^2, becomes the new leading term to be divided in the next iteration.

  4. Bring Down: Bring down the next term from the dividend (-8x). Now we have -14x^2 - 8x. Bringing down the next term is a straightforward step, but it's crucial for maintaining the flow of the long division process. It's like adding another digit to the remainder in regular long division. The term you bring down becomes part of the new polynomial that you'll be dividing in the next step. This step ensures that you're working with the entire dividend, one term at a time, until you've accounted for all the terms. In this case, bringing down the -8x term creates the polynomial -14x^2 - 8x, which will be the focus of the next iteration of the division. It's important to note that you bring down the term along with its sign, ensuring that you maintain the correct algebraic expression.

  5. Repeat: Repeat steps 1-4 with the new polynomial (-14x^2 - 8x). So, we divide -14x^2 by x, which gives us -14x. This is the next term in our quotient. The repetition of steps 1-4 is the heart of the long division process. It's an iterative procedure that you continue until you've brought down all the terms from the dividend and either reached a remainder of zero or a remainder that has a lower degree than the divisor. Each iteration refines the quotient and reduces the complexity of the remaining polynomial. In this step, we divide the leading term of the new polynomial, -14x^2, by the leading term of the divisor, x, resulting in -14x. This becomes the next term in the quotient. The process then continues with multiplying the divisor by -14x, subtracting the result from the current polynomial, and bringing down the next term, if there is one. This cyclical nature of the long division algorithm makes it a systematic and reliable method for dividing polynomials of any degree.

  6. Multiply: Multiply the divisor (x - 2) by -14x. -14x * (x - 2) = -14x^2 + 28x. This multiplication step is similar to the previous one, but now we're working with the term -14x from the quotient. It's crucial to pay attention to the signs during this step, as multiplying by a negative term can be tricky. The result, -14x^2 + 28x, will be subtracted from the current polynomial in the next step. This step helps you account for the portion of the polynomial -14x^2 - 8x that is divisible by (x - 2). It's a critical part of the process that ensures you're accurately determining the quotient and the remainder. Moreover, this multiplication step reinforces the distributive property of multiplication over addition and subtraction, which is a fundamental concept in algebra.

  7. Subtract: Subtract the result (-14x^2 + 28x) from the current polynomial (-14x^2 - 8x). (-14x^2 - 8x) - (-14x^2 + 28x) = -36x. Subtraction is a crucial step in long division, and it requires careful attention to signs. In this case, we are subtracting (-14x^2 + 28x) from (-14x^2 - 8x). Remember to distribute the negative sign when subtracting, which means changing the signs of the terms being subtracted and then combining like terms. This step can be prone to errors, so it's a good idea to double-check your work. The result of the subtraction, -36x, becomes the new leading term to be divided in the next iteration. This step essentially refines the remainder and sets the stage for continuing the division process. A clear understanding of how to subtract polynomials is essential for mastering long division.

  8. Bring Down: Bring down the last term from the dividend (-13). Now we have -36x - 13. Bringing down the last term from the dividend signals that we are nearing the end of the long division process. It's a simple step, but it's crucial for ensuring that we account for all the terms in the dividend. The term we bring down, -13, joins the current polynomial, -36x, to form the new polynomial -36x - 13. This polynomial will be the final one we divide using the divisor. It's like completing the last part of a puzzle, where bringing down the final piece allows us to see the complete picture. This step sets up the final iteration of the division, where we will determine the last term of the quotient and the final remainder.

  9. Repeat: Repeat steps 1-4 with the new polynomial (-36x - 13). Divide -36x by x, which gives us -36. This is the last term in our quotient. Repeating the division process with the polynomial -36x - 13 is the final step in determining the quotient and the remainder. We divide the leading term, -36x, by the leading term of the divisor, x, which gives us -36. This becomes the last term in our quotient. It's like the final piece of the puzzle falling into place, completing the quotient. This step also highlights the iterative nature of long division, where we repeatedly apply the same set of steps until we reach a remainder that is either zero or of a lower degree than the divisor. A clear understanding of this iterative process is key to mastering long division and applying it to more complex problems.

  10. Multiply: Multiply the divisor (x - 2) by -36. -36 * (x - 2) = -36x + 72. Multiplying the divisor (x - 2) by the last term of the quotient, -36, is a critical step in the final iteration of the division process. This step ensures that we account for the portion of the polynomial -36x - 13 that is divisible by the divisor. The result, -36x + 72, will be subtracted from the current polynomial in the next step to determine the remainder. This multiplication step also reinforces the distributive property of multiplication over addition and subtraction, a fundamental concept in algebra. Paying close attention to the signs during this multiplication is essential to avoid errors and ensure an accurate result.

  11. Subtract: Subtract the result (-36x + 72) from the current polynomial (-36x - 13). (-36x - 13) - (-36x + 72) = -85. Subtracting the result (-36x + 72) from the current polynomial (-36x - 13) is the final subtraction in the long division process. This step determines the remainder, which is the part of the dividend that is not evenly divisible by the divisor. Remember to distribute the negative sign when subtracting, changing the signs of the terms being subtracted and then combining like terms. The result of this subtraction is -85, which is the remainder. Since the degree of the remainder (which is a constant) is less than the degree of the divisor (which is 1), we have completed the division. This step is the culmination of the iterative process and provides the final piece of information needed to express the result of the division.

The Result

So, the quotient is x^2 - 14x - 36, and the remainder is -85. We write the final answer as:

(x^3 - 16x^2 - 8x - 13) á (x - 2) = x^2 - 14x - 36 - 85/(x - 2)

Remember, the remainder is written as a fraction with the divisor as the denominator. Expressing the result with the remainder as a fraction is a standard practice in polynomial division. It provides a complete picture of the division, showing both the quotient (the result of the division) and the remainder (the part that is left over). In this case, the remainder is -85, and the divisor is (x - 2), so we write the remainder as -85/(x - 2). This fraction is added (or subtracted, depending on the sign) to the quotient to give the final result. It's important to include the remainder in the final answer because it represents the portion of the dividend that was not evenly divisible by the divisor. This comprehensive representation of the division is not only mathematically accurate but also provides valuable information about the relationship between the polynomials involved. For example, it can help in identifying asymptotes and other key features of rational functions.

Tips and Tricks

  • Stay Organized: Keep your terms lined up. This will save you from making silly mistakes.
  • Double-Check: Always double-check your multiplication and subtraction.
  • Placeholders: Don't forget to use placeholders for missing terms.

Common Mistakes

  • Sign Errors: Watch out for those negative signs!
  • Skipping Terms: Make sure you bring down all the terms.
  • Incorrect Multiplication: Double-check your multiplication steps.

Polynomial division might seem tricky at first, but with practice, you'll get the hang of it. Just remember to take it one step at a time, stay organized, and don't be afraid to double-check your work. You've got this! And that's a wrap on polynomial division, guys. Keep practicing, and you'll be a pro in no time!

Polynomial division can feel like navigating a maze at first, but with the right strategies and a bit of practice, it becomes a straightforward process. Let's delve deeper into some advanced techniques, common pitfalls, and practical tips that will help you master polynomial division. Whether you're a student tackling algebra or someone brushing up on their math skills, this comprehensive guide will equip you with the knowledge and confidence to divide polynomials like a pro.

Advanced Techniques for Polynomial Division

Synthetic Division: A Speedier Approach

Synthetic division is a streamlined method for dividing a polynomial by a linear divisor of the form x - a. It's faster and more compact than long division, making it a favorite for quick problem-solving. To use synthetic division, you only work with the coefficients of the polynomial, which simplifies the process. For example, to divide x^3 - 16x^2 - 8x - 13 by x - 2, you would set up the synthetic division table with the coefficients 1, -16, -8, and -13, and the value 2 (from x - 2). The process involves bringing down the first coefficient, multiplying it by the divisor's root, adding it to the next coefficient, and repeating until all coefficients have been processed. The final numbers in the table represent the coefficients of the quotient and the remainder. Synthetic division is particularly useful when dealing with problems that involve repeated divisions or when the divisor is a simple linear expression. However, it's important to remember that synthetic division can only be used when dividing by a linear divisor, while long division is applicable to divisors of any degree. Therefore, understanding both methods is crucial for tackling a wide range of polynomial division problems.

The Remainder Theorem: A Quick Check

The Remainder Theorem is a handy tool that allows you to find the remainder of a polynomial division without actually performing the division. It states that if you divide a polynomial f(x) by x - a, the remainder is f(a). For instance, to find the remainder when x^3 - 16x^2 - 8x - 13 is divided by x - 2, you simply evaluate the polynomial at x = 2. This gives you (2)^3 - 16(2)^2 - 8(2) - 13 = 8 - 64 - 16 - 13 = -85, which matches the remainder we found using long division. The Remainder Theorem is not only a quick way to check your work but also a valuable concept in its own right. It forms the basis for many algebraic manipulations and is closely related to the Factor Theorem, which states that x - a is a factor of f(x) if and only if f(a) = 0. Together, these theorems provide powerful tools for factoring polynomials, finding roots, and simplifying algebraic expressions. Moreover, the Remainder Theorem has applications in numerical analysis, where it is used to approximate the values of polynomials and their derivatives. Therefore, mastering the Remainder Theorem is an essential step in developing a deeper understanding of polynomial behavior and algebraic techniques.

Common Mistakes and How to Avoid Them

Sign Slip-Ups

One of the most frequent errors in polynomial division is making mistakes with signs, especially during subtraction. To avoid this, always distribute the negative sign carefully when subtracting polynomials. It's a good practice to write out the subtraction step explicitly, changing the signs of the terms being subtracted, before combining like terms. For example, when subtracting (x^3 - 2x^2) from (x^3 - 16x^2), rewrite it as x^3 - 16x^2 - x^3 + 2x^2 before simplifying. This visual reminder helps prevent sign errors. Sign errors can easily propagate through the entire division process, leading to an incorrect result. Therefore, taking the time to be meticulous with signs is a worthwhile investment. Moreover, using different colored pens or highlighters can help distinguish between terms and their signs, making it easier to keep track of the operations. Developing a habit of double-checking each step, particularly the subtraction steps, will significantly reduce the likelihood of sign-related mistakes. A clear understanding of the rules of sign manipulation is also crucial for mastering polynomial division and other algebraic operations.

Missing Placeholders

Forgetting to include placeholders for missing terms in the dividend can throw off the entire division process. Always ensure that your polynomial includes terms for every degree, from the highest power down to the constant term. If a term is missing, insert a placeholder with a coefficient of zero. For example, if you're dividing x^3 - 8x - 13, rewrite it as x^3 + 0x^2 - 8x - 13. These placeholders maintain the correct alignment of terms and ensure that the division algorithm works correctly. Without placeholders, terms may not line up properly, leading to incorrect subtractions and a flawed quotient. This is particularly important when using long division, where the visual alignment of terms is critical for accurate calculations. Missing placeholders can also lead to confusion when interpreting the final result, as the coefficients of the quotient may be assigned to the wrong powers of the variable. Therefore, always scan the dividend for missing terms and insert placeholders as needed before beginning the division process. This simple step can prevent a significant amount of errors and ensure a smooth and accurate division.

Misalignment of Terms

Keeping terms properly aligned is essential for accurate polynomial division. Make sure that like terms (terms with the same power of x) are lined up vertically during the division process. This makes it easier to combine them correctly during subtraction. Use columns to organize your work, and don't hesitate to rewrite the problem if your terms start to drift out of alignment. Misalignment of terms can lead to errors in subtraction and addition, resulting in an incorrect quotient and remainder. This is particularly true when dealing with polynomials of higher degrees, where the number of terms can make it challenging to keep everything organized. A systematic approach to setting up the division problem, with clear columns for each power of x, can help prevent misalignment. Moreover, using graph paper or lined paper can provide a visual aid for maintaining alignment. It's also a good practice to periodically check the alignment of terms as you work through the division process, making adjustments as needed. By prioritizing organization and alignment, you can significantly reduce the risk of errors and improve the accuracy of your polynomial division.

Practical Tips for Mastering Polynomial Division

Practice Makes Perfect

The best way to master polynomial division is through practice. Work through a variety of problems, starting with simpler examples and gradually moving on to more complex ones. The more you practice, the more comfortable you'll become with the process, and the easier it will be to spot potential errors. Practice also helps you develop a better understanding of the underlying concepts and the relationships between polynomials, divisors, quotients, and remainders. Start with problems that involve dividing by linear divisors, as these are simpler and allow you to focus on the mechanics of the division process. Then, move on to problems with quadratic or higher-degree divisors, which require a more careful application of the long division algorithm. Working through a mix of problems, including those with missing terms and negative coefficients, will help you develop a well-rounded skill set. Don't hesitate to seek out additional practice problems from textbooks, online resources, or worksheets. The key is to consistently challenge yourself and reinforce your understanding through repetition.

Break It Down

If you're struggling with a particularly challenging problem, try breaking it down into smaller steps. Focus on one step at a time, and make sure you understand each step before moving on to the next. This can make the process feel less overwhelming and help you identify any areas where you might be making mistakes. Polynomial division involves a series of interconnected steps, and mastering each step individually is crucial for successful problem-solving. Start by focusing on setting up the problem correctly, ensuring that the dividend and divisor are in the correct order and that any missing terms are accounted for with placeholders. Then, practice the division, multiplication, and subtraction steps separately, paying close attention to the signs and alignment of terms. If you encounter a mistake, take the time to analyze where it occurred and why. Breaking down the problem also allows you to focus on the underlying logic of the division process, which can lead to a deeper understanding and improved retention. By approaching polynomial division in a step-by-step manner, you can build confidence and develop a systematic approach to solving even the most complex problems.

Check Your Work

Always check your work after completing a polynomial division problem. You can do this by multiplying the quotient by the divisor and adding the remainder. The result should be the original dividend. This is a great way to catch any mistakes you might have made during the division process. Checking your work not only ensures accuracy but also reinforces your understanding of the relationship between the dividend, divisor, quotient, and remainder. The verification process involves the reverse of division, allowing you to confirm that the division was performed correctly. If the product of the quotient and divisor, plus the remainder, does not equal the dividend, then you know there is an error in your calculations. Take the time to review each step of the division process to identify and correct the mistake. This practice of self-checking is a valuable skill that can be applied to many areas of mathematics. It promotes carefulness, accuracy, and a deeper understanding of the underlying concepts. Moreover, checking your work can build confidence in your problem-solving abilities and help you develop a more systematic approach to tackling mathematical challenges.

By following these tips and practicing regularly, you'll be well on your way to mastering polynomial division. Remember, it's a fundamental skill that will serve you well in many areas of mathematics and beyond. So, embrace the challenge, and enjoy the journey of learning this powerful algebraic technique!

In this part of the guide, we will solve the specific polynomial division problem you provided: (x^3 - 16x^2 - 8x - 13) á (x - 2). We will walk through each step of the long division process in detail, reinforcing the concepts and techniques we've discussed so far. By following along with this example, you'll gain a clearer understanding of how to apply the long division algorithm and how to avoid common mistakes. This step-by-step solution will serve as a practical demonstration of the principles and tips we've covered, helping you build confidence in your ability to tackle similar problems.

Setting Up the Long Division

  1. Write the dividend (x^3 - 16x^2 - 8x - 13) inside the division symbol and the divisor (x - 2) outside. Ensure the dividend is in descending order of powers of x, and include placeholders if necessary. In this case, the dividend is already in the correct order, and there are no missing terms, so we can proceed directly to the division process.

Step-by-Step Division Process

  1. Divide the First Terms: Divide the first term of the dividend (x^3) by the first term of the divisor (x). x^3 á x = x^2. Write x^2 as the first term of the quotient above the division symbol. This step sets the stage for the entire division process, determining the initial term of the quotient that will be used to systematically reduce the dividend. The result of this division, x^2, represents the portion of the dividend that is accounted for by the divisor in this first iteration.

  2. Multiply: Multiply the entire divisor (x - 2) by the first term of the quotient (x^2). x^2 * (x - 2) = x^3 - 2x^2. Write the result below the dividend, aligning like terms. This multiplication step is the reverse of the initial division step, allowing us to determine how much of the dividend is divisible by the divisor based on the first term of the quotient. The result, x^3 - 2x^2, is what we will subtract from the dividend in the next step.

  3. Subtract: Subtract the result (x^3 - 2x^2) from the corresponding terms in the dividend (x^3 - 16x^2). (x^3 - 16x^2) - (x^3 - 2x^2) = -14x^2. Be careful to distribute the negative sign correctly. Write the result below. This subtraction step is crucial for determining the remaining portion of the dividend that needs to be divided. It involves changing the signs of the terms being subtracted and then combining like terms. The result, -14x^2, becomes the leading term of the new polynomial that will be used in the next iteration of the division process.

  4. Bring Down: Bring down the next term from the dividend (-8x). Now we have -14x^2 - 8x. Write this below the result from the subtraction. Bringing down the next term from the dividend is a straightforward step that ensures we account for all terms in the dividend. The term we bring down, -8x, joins the result of the previous subtraction, -14x^2, to form the new polynomial -14x^2 - 8x. This polynomial will be the focus of the next iteration of the division process.

  5. Repeat: Repeat steps 1-4 with the new polynomial (-14x^2 - 8x). Divide -14x^2 by x, which gives us -14x. Write -14x as the next term in the quotient. This repetition of the division process is the heart of the long division algorithm. We divide the leading term of the new polynomial, -14x^2, by the leading term of the divisor, x, resulting in -14x. This becomes the next term in the quotient, and the process continues with multiplying the divisor by -14x, subtracting the result from the current polynomial, and bringing down the next term, if there is one.

  6. Multiply: Multiply the divisor (x - 2) by -14x. -14x * (x - 2) = -14x^2 + 28x. Write the result below the current polynomial, aligning like terms. This multiplication step ensures we account for the portion of the polynomial -14x^2 - 8x that is divisible by the divisor, based on the term -14x in the quotient. The result, -14x^2 + 28x, will be subtracted from the current polynomial in the next step.

  7. Subtract: Subtract the result (-14x^2 + 28x) from the current polynomial (-14x^2 - 8x). (-14x^2 - 8x) - (-14x^2 + 28x) = -36x. Remember to distribute the negative sign. Write the result below. This subtraction step refines the remainder and sets the stage for the final iteration of the division process. We subtract (-14x^2 + 28x) from (-14x^2 - 8x), carefully changing the signs of the terms being subtracted and then combining like terms. The result, -36x, becomes the new leading term to be divided in the next step.

  8. Bring Down: Bring down the last term from the dividend (-13). Now we have -36x - 13. Write this below the result from the subtraction. Bringing down the last term from the dividend signals that we are nearing the end of the long division process. The term we bring down, -13, joins the current polynomial, -36x, to form the new polynomial -36x - 13. This polynomial will be the final one we divide using the divisor.

  9. Repeat: Repeat steps 1-4 with the new polynomial (-36x - 13). Divide -36x by x, which gives us -36. Write -36 as the last term in the quotient. This final repetition of the division process determines the last term of the quotient and the remainder. We divide the leading term, -36x, by the leading term of the divisor, x, resulting in -36. This completes the quotient, and we move on to the final multiplication and subtraction steps to determine the remainder.

  10. Multiply: Multiply the divisor (x - 2) by -36. -36 * (x - 2) = -36x + 72. Write the result below the current polynomial, aligning like terms. This multiplication step ensures we account for the remaining portion of the polynomial -36x - 13 that is divisible by the divisor. The result, -36x + 72, will be subtracted from the current polynomial in the final subtraction step to determine the remainder.

  11. Subtract: Subtract the result (-36x + 72) from the current polynomial (-36x - 13). (-36x - 13) - (-36x + 72) = -85. Write the result below. This final subtraction step determines the remainder, which is the part of the dividend that is not evenly divisible by the divisor. The result, -85, is the remainder in this case.

Writing the Result

The quotient is x^2 - 14x - 36, and the remainder is -85. We write the final answer as:

(x^3 - 16x^2 - 8x - 13) á (x - 2) = x^2 - 14x - 36 - 85/(x - 2)

This complete step-by-step solution demonstrates how to apply the long division algorithm to divide polynomials. By carefully following each step and paying attention to the details, you can successfully solve polynomial division problems and master this essential algebraic technique.

So, there you have it, guys! We've tackled polynomial division head-on, breaking down the steps, sharing tips and tricks, and working through a real example. Polynomial division might seem daunting at first, but with a clear understanding of the process and plenty of practice, you'll be simplifying complex expressions like a mathematical maestro in no time. Remember to stay organized, double-check your work, and don't hesitate to break down the problem into smaller, more manageable steps. And if you ever get stuck, just revisit this guide, and you'll be back on track in no time. Keep practicing, keep learning, and most importantly, keep having fun with math! Polynomial division is not just a skill; it's a tool that opens doors to more advanced mathematical concepts and real-world applications. From calculus to engineering, the ability to divide polynomials efficiently and accurately is invaluable. So, embrace the challenge, hone your skills, and unleash the power of polynomial division in your mathematical endeavors. You've got this!