Solving Polynomial Division 5a⁴ + A² - 2a - 3 By A - 1 A Step-by-Step Guide

October 14, 2025 by Scholario Team 76 views

Hey guys! Are you struggling with polynomial division? No worries, we've all been there! Polynomial division might seem intimidating at first, but trust me, once you break it down step by step, it becomes much more manageable. Let's dive into how you can tackle the problem (5a⁴ + a² - 2a - 3) ÷ (a - 1) and get it done before tomorrow! This guide will walk you through each stage, ensuring you not only get the answer but also understand the process.

Understanding Polynomial Division

Before we jump into the solution, let's understand polynomial division. Think of it as the long division you learned in elementary school, but now we're dealing with algebraic expressions instead of numbers. The goal is the same: to find out how many times one polynomial (the divisor) fits into another polynomial (the dividend). In our case, (5a⁴ + a² - 2a - 3) is the dividend, and (a - 1) is the divisor.

The main keywords here are polynomial division, dividend, and divisor. Polynomial division is the process we are undertaking, the dividend is the polynomial being divided (5a⁴ + a² - 2a - 3), and the divisor is the polynomial we are dividing by (a - 1). Remembering these terms will help you follow along more easily. The goal of polynomial division, much like regular long division, is to find out how many times the divisor fits into the dividend. Mastering this concept opens doors to simplifying complex algebraic expressions and solving intricate equations. So, let's break it down further. We will proceed step by step to ensure clarity and comprehension.

Polynomial division is not just a mathematical procedure; it’s a foundational skill in algebra. Understanding it allows you to simplify complex expressions, solve equations, and even tackle more advanced topics like calculus. It’s like learning the alphabet before writing a novel – you need the basics to build on. So, let’s treat this as an essential tool in your mathematical toolkit. As we move forward, remember that each step builds on the previous one. Don't rush the process; take your time to understand each action. Polynomial division might seem daunting, but with the right approach, it becomes a manageable and even interesting problem to solve. Let’s start by setting up the problem correctly.

Step-by-Step Solution

1. Set Up the Division

First, we'll set up the long division format. Write the dividend (5a⁴ + a² - 2a - 3) inside the division symbol and the divisor (a - 1) outside. Make sure to include placeholders for any missing terms. Notice that we're missing a term for a³, so we'll add 0a³ to keep everything aligned:

        _____________
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3

The setup is crucial because it organizes the problem, making it easier to follow each step. Think of it as preparing your workspace before starting a project. A well-organized setup prevents errors and makes the entire process smoother. The placeholders, like 0a³, are essential because they maintain the correct alignment of terms, ensuring that you add and subtract like terms accurately. Without these placeholders, you might mix up the powers of 'a', leading to an incorrect result. Setting up the division correctly is like laying a solid foundation for a building – it supports everything that comes next.

Moreover, the long division format visually breaks down the problem into smaller, manageable parts. This helps in focusing on one step at a time, reducing the cognitive load. Each step involves a simple process: divide, multiply, subtract, and bring down. By organizing the problem in this format, we are essentially creating a roadmap that guides us to the solution. So, take a moment to ensure your setup is perfect. It’s the key to solving the polynomial division accurately and efficiently. The next step involves dividing the first term of the dividend by the first term of the divisor, which will give us the first term of the quotient. Let’s move on to that now.

2. Divide the First Terms

Now, divide the first term of the dividend (5a⁴) by the first term of the divisor (a). 5a⁴ ÷ a = 5a³. This is the first term of our quotient. Write it above the division symbol, aligning it with the a⁴ term:

        5a³__________
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3

Dividing the first terms sets the stage for the rest of the division process. It's like the opening move in a chess game – it dictates the direction of the game. By focusing on the leading terms, we simplify the problem into smaller, more manageable parts. Remember, the goal here is to find the term that, when multiplied by the divisor, will eliminate the leading term of the dividend. This is a crucial step in reducing the complexity of the polynomial. Getting this first division right is essential, as it directly influences the accuracy of the subsequent steps.

Moreover, this step highlights the importance of understanding the rules of exponents. When dividing terms with exponents, you subtract the exponents. For example, a⁴ divided by a (which is a¹) is a³ (4 - 1 = 3). This basic rule is fundamental to polynomial division. So, ensure you are comfortable with these rules before proceeding. Think of this step as setting a puzzle piece in place; it’s the first piece, and it’s essential to get it right for the rest of the puzzle to fit together. Once we have the first term of the quotient, we move on to multiplying this term by the entire divisor. Let’s see how that works in the next step.

3. Multiply and Subtract

Multiply 5a³ by the entire divisor (a - 1): 5a³ * (a - 1) = 5a⁴ - 5a³. Write this result below the dividend and subtract it:

        5a³__________
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)

Subtract: (5a⁴ + 0a³) - (5a⁴ - 5a³) = 5a³

        5a³__________
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³

Multiplication and subtraction are key operations in long division, both with numbers and polynomials. Multiplying 5a³ by (a - 1) helps us determine the part of the dividend that can be 'canceled out' by the divisor. Think of it as figuring out how much of the bigger number can be neatly divided by the smaller one. The subtraction step is then crucial because it shows us what's left over after this division. In our case, we subtract (5a⁴ - 5a³) from (5a⁴ + 0a³) to find the remainder, which is 5a³. This remainder is what we'll continue to work with in the next steps. Accurate multiplication and subtraction are vital to prevent errors from accumulating as we proceed through the problem.

Moreover, paying close attention to signs during subtraction is super important. It's a common area for mistakes! Remember that subtracting a negative is the same as adding a positive. For example, subtracting -5a³ means we're actually adding 5a³. Getting the signs right ensures that we correctly determine the new polynomial to work with. This step is like balancing an equation – we need to ensure that both sides remain equal as we perform operations. Once we've subtracted correctly, we bring down the next term from the dividend, setting us up for the next round of division. So, let’s bring down the next term and see what happens.

4. Bring Down the Next Term

Bring down the next term from the dividend (+a²):

        5a³__________
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²

Bringing down the next term is a simple but crucial step. It's like adding another piece to the puzzle. We've dealt with the 5a⁴ term, and now we're moving on to the next part of the dividend. This step keeps the process going and ensures we consider all parts of the polynomial. Think of it as moving to the next digit in a regular long division problem. Bringing down the term means we now have a new polynomial to work with: 5a³ + a². This new polynomial is what we'll divide by our divisor (a - 1) in the next iteration of the process. It's a continuous cycle of dividing, multiplying, subtracting, and bringing down until we've used all the terms in the dividend.

Moreover, this step highlights the systematic nature of polynomial division. It’s a repetitive process, which means once you understand the basic steps, you can apply them again and again. This repetition is what makes the process manageable. Each time we bring down a term, we’re essentially resetting the problem, but with a smaller polynomial. This makes the overall division more approachable. Think of it as breaking a big task into smaller, more achievable subtasks. So, now that we've brought down the next term, we're ready to repeat the division process. Let’s divide the new leading term by the leading term of the divisor and see what we get.

5. Repeat the Process

Repeat the division process: Divide the first term of the new polynomial (5a³) by the first term of the divisor (a). 5a³ ÷ a = 5a². Add this to the quotient:

        5a³ + 5a²______
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²

Multiply 5a² by (a - 1): 5a² * (a - 1) = 5a³ - 5a². Write this below and subtract:

        5a³ + 5a²______
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)

Subtract: (5a³ + a²) - (5a³ - 5a²) = 6a²

        5a³ + 5a²______
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)
        _________
                    6a²

Repeating the process is where the rhythm of polynomial division really starts to become clear. It’s like following a recipe – once you’ve done the first few steps, you understand the pattern and can continue more confidently. Dividing 5a³ by a gives us 5a², which we add to our quotient. This new term is crucial in getting us closer to the final answer. Then, multiplying 5a² by the divisor (a - 1) and subtracting it from our current polynomial (5a³ + a²) helps us reduce the polynomial further. This cycle of divide, multiply, and subtract is the heart of polynomial division.

Moreover, each repetition reinforces your understanding of the process. You’re not just following steps; you’re internalizing how the division works. Paying attention to the signs during subtraction remains important here. Subtracting (5a³ - 5a²) means we're doing 5a³ + a² - 5a³ + 5a², which simplifies to 6a². Getting the signs right is like tuning an instrument – it ensures the harmony of the result. After the subtraction, we bring down the next term again. So, let's bring down the -2a and continue the process.

6. Continue Repeating

Bring down the next term (-2a):

        5a³ + 5a²______
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)
        _________
                    6a² - 2a

Divide 6a² by a: 6a² ÷ a = 6a. Add this to the quotient:

        5a³ + 5a² + 6a____
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)
        _________
                    6a² - 2a

Multiply 6a by (a - 1): 6a * (a - 1) = 6a² - 6a. Write this below and subtract:

        5a³ + 5a² + 6a____
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)
        _________
                    6a² - 2a
       -(6a² - 6a)

Subtract: (6a² - 2a) - (6a² - 6a) = 4a

        5a³ + 5a² + 6a____
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)
        _________
                    6a² - 2a
       -(6a² - 6a)
        _________
                          4a

Continuing the repetition brings us closer to the final result. We've brought down -2a, divided 6a² by a to get 6a, and added 6a to the quotient. This consistent process is like knitting a sweater – each loop builds on the previous one, gradually creating the whole fabric. Multiplying 6a by (a - 1) and subtracting helps us to further reduce the polynomial. Remember, the goal is to keep reducing the degree of the polynomial until we can no longer divide evenly by the divisor. This repetitive cycle is what makes the algorithm so effective.

Moreover, this step further ingrains the importance of methodical work. Polynomial division is not about doing everything at once; it's about taking small, manageable steps. Each step, from bringing down the term to performing the subtraction, is a building block. Accuracy in each step is crucial for the final result. Think of this as climbing a staircase – each step you take brings you closer to the top. We’re almost there! Let’s bring down the final term and complete the division.

7. Final Steps

Bring down the last term (-3):

        5a³ + 5a² + 6a____
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)
        _________
                    6a² - 2a
       -(6a² - 6a)
        _________
                          4a - 3

Divide 4a by a: 4a ÷ a = 4. Add this to the quotient:

        5a³ + 5a² + 6a + 4
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)
        _________
                    6a² - 2a
       -(6a² - 6a)
        _________
                          4a - 3

Multiply 4 by (a - 1): 4 * (a - 1) = 4a - 4. Write this below and subtract:

        5a³ + 5a² + 6a + 4
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)
        _________
                    6a² - 2a
       -(6a² - 6a)
        _________
                          4a - 3
       -(4a - 4)

Subtract: (4a - 3) - (4a - 4) = 1

        5a³ + 5a² + 6a + 4
a - 1 | 5a⁴ + 0a³ + a² - 2a - 3
       -(5a⁴ - 5a³)
        _________
              5a³ + a²
       -(5a³ - 5a²)
        _________
                    6a² - 2a
       -(6a² - 6a)
        _________
                          4a - 3
       -(4a - 4)
        _________
                              1

Reaching the final steps is like arriving at the summit of a mountain – you’ve put in the effort, and now you can see the result. Bringing down the last term (-3) sets us up for the final division. Dividing 4a by a gives us 4, which we add to our quotient. This is the last piece of the quotient we need to find. Multiplying 4 by the divisor (a - 1) and subtracting allows us to find the remainder. The remainder is what’s left over after we've divided as much as we can. It’s an essential part of the answer.

Moreover, this final subtraction highlights the complete division process. We started with a complex polynomial and systematically reduced it until we couldn’t divide any further. The remainder, in this case 1, tells us that (a - 1) doesn't divide perfectly into (5a⁴ + a² - 2a - 3), but we've found how close it gets. Think of this as fitting puzzle pieces together – if there's a remainder, it means the pieces don't fit perfectly, but we know how much is left over. Now, let’s write out the final answer, including the quotient and the remainder.

Final Answer

The quotient is 5a³ + 5a² + 6a + 4, and the remainder is 1. So, the final answer is:

5a³ + 5a² + 6a + 4 + 1/(a - 1)

And that's it! You've successfully divided the polynomial. Remember, practice makes perfect, so try a few more examples to get the hang of it. You got this!

Understanding the final answer in polynomial division is like understanding the complete picture after assembling a puzzle. The quotient, 5a³ + 5a² + 6a + 4, represents the part of the dividend that the divisor can perfectly divide into. Think of it as the main part of the division result. The remainder, 1, is what’s left over – the part that couldn’t be evenly divided. To express the complete answer, we write the quotient plus the remainder over the divisor, which gives us 5a³ + 5a² + 6a + 4 + 1/(a - 1).

Moreover, writing the answer in this format ensures that we’ve accounted for every part of the division. The quotient and remainder together give us a full and accurate solution. This is akin to expressing a fraction as a mixed number – you have the whole number part (the quotient) and the fractional part (the remainder over the divisor). Understanding how to interpret and write the final answer is just as important as the division process itself. It demonstrates that you comprehend not only the mechanics of the division but also what the result means. So, take pride in reaching this final step – you’ve successfully navigated a challenging problem! If you have time, double-check your work to make sure everything is correct. And remember, practice is key to mastering polynomial division.

Tips for Success

Stay Organized: Keep your work neat and aligned to avoid errors.
Double-Check Signs: Pay close attention to positive and negative signs during subtraction.
Practice Regularly: The more you practice, the easier it becomes.

Conclusion

Polynomial division can seem tough, but with a systematic approach, you can conquer it! Remember to take it step by step, and don't hesitate to ask for help if you get stuck. Good luck, and get that assignment done! You've got this! This comprehensive guide should help you not only solve the problem but also understand the process behind it. Keep practicing, and you'll become a polynomial division pro in no time! Remember, the key is to break down the problem into manageable steps and stay organized. You've got the tools now – go ace that assignment!