Hey guys! Today, we're diving into the world of polynomial division. It might sound intimidating, but trust me, it's like solving a puzzle! We're going to break down the problem (x³ + 6x² - 34x + 61) ÷ (x + 10) step by step, making it super easy to understand. Polynomial division is a crucial skill in algebra, popping up in various areas of mathematics and even in real-world applications like engineering and computer graphics. So, let's get started and conquer this topic together!
Understanding Polynomial Division
Before we jump into the problem, let's quickly recap what polynomial division actually means. Think of it like regular long division but with algebraic expressions. Polynomial division helps us break down complex polynomials into simpler parts, which is incredibly useful for solving equations, factoring, and simplifying expressions. We're essentially trying to find out how many times the divisor (in our case, x + 10) fits into the dividend (x³ + 6x² - 34x + 61). The result we get is called the quotient, and sometimes there's a little bit left over, which we call the remainder. This remainder is a crucial part of the answer, as it tells us how much of the dividend wasn't perfectly divisible by the divisor. Mastering this concept is essential for any algebra student, as it lays the groundwork for more advanced topics like synthetic division and solving rational equations.
Long Division Method
We'll be using the long division method, which is a classic and reliable way to tackle these problems. The long division method is a systematic approach that allows us to divide polynomials of any degree. It's like following a recipe – if you stick to the steps, you'll get the right result. This method involves a series of steps: dividing, multiplying, subtracting, and bringing down. We repeat these steps until we've divided all the terms of the dividend. The quotient is formed by the terms we obtain in the division process, and any leftover is the remainder. Long division provides a clear visual representation of the division process, making it easier to track each step and avoid errors. It's a foundational technique that every student should be comfortable with, as it provides a solid understanding of how polynomial division works.
Step-by-Step Solution
Okay, let's get our hands dirty and solve (x³ + 6x² - 34x + 61) ÷ (x + 10). We'll walk through each step carefully.
Step 1: Set up the Long Division
First, we set up the problem just like regular long division. We write the dividend (x³ + 6x² - 34x + 61) inside the division bracket and the divisor (x + 10) outside. Make sure the polynomials are written in descending order of their exponents – this helps keep things organized and prevents confusion. Setting up the problem correctly is the first crucial step in long division. A neat and organized setup makes it easier to follow the steps and minimize errors. This visual representation helps us keep track of the terms and the division process, ensuring we don't miss any steps or make mistakes in the calculations.
________________________
x + 10 | x³ + 6x² - 34x + 61
Step 2: Divide the Leading Terms
Now, we focus on the leading terms. We divide the leading term of the dividend (x³) by the leading term of the divisor (x). x³ divided by x is x². This x² is the first term of our quotient, so we write it above the division bracket, aligned with the x² term in the dividend. Dividing the leading terms is the key to starting the long division process. It determines the first term of the quotient and sets the stage for the subsequent steps. By focusing on the leading terms, we systematically reduce the degree of the dividend until we arrive at the remainder.
x²____________________
x + 10 | x³ + 6x² - 34x + 61
Step 3: Multiply the Quotient Term by the Divisor
Next, we multiply the x² (the first term of our quotient) by the entire divisor (x + 10). x² times (x + 10) equals x³ + 10x². We write this result below the dividend, aligning like terms. Multiplying the quotient term by the divisor is a crucial step in the long division process. It allows us to determine how much of the dividend is accounted for by the current term of the quotient. This step sets up the subtraction that follows, which reduces the dividend and brings us closer to the final answer. Accurate multiplication is essential to ensure the correctness of the subsequent steps.
x²____________________
x + 10 | x³ + 6x² - 34x + 61
x³ + 10x²
Step 4: Subtract and Bring Down
Now, we subtract (x³ + 10x²) from (x³ + 6x²). This gives us x³ + 6x² - (x³ + 10x²) = -4x². Then, we bring down the next term from the dividend, which is -34x. So, we now have -4x² - 34x. Subtracting and bringing down are key steps in the long division process. Subtraction allows us to eliminate terms from the dividend and reduce its degree. Bringing down the next term prepares us for the next iteration of the division process. These steps are repeated until we have divided all the terms of the dividend.
x²____________________
x + 10 | x³ + 6x² - 34x + 61
x³ + 10x²
-----------
-4x² - 34x
Step 5: Repeat the Process
We repeat the process with the new expression, -4x² - 34x. We divide the leading term -4x² by the leading term of the divisor x, which gives us -4x. This is the next term of our quotient, so we write it next to x². Then, we multiply -4x by (x + 10) to get -4x² - 40x, and we write this below -4x² - 34x. Repeating the process is the heart of long division. We continue dividing, multiplying, and subtracting until the degree of the remaining dividend is less than the degree of the divisor. Each iteration of the process brings us closer to the final quotient and remainder. This repetitive nature makes long division a systematic and reliable method for dividing polynomials.
x² - 4x____________
x + 10 | x³ + 6x² - 34x + 61
x³ + 10x²
-----------
-4x² - 34x
-4x² - 40x
Step 6: Subtract Again and Bring Down
Subtracting -4x² - 40x from -4x² - 34x gives us -4x² - 34x - (-4x² - 40x) = 6x. We bring down the last term from the dividend, which is +61. So, we have 6x + 61. This step is crucial for continuing the division process. Subtracting the appropriate terms allows us to reduce the dividend further, and bringing down the next term sets up the next iteration. The careful execution of this step ensures that we account for all terms in the dividend and accurately determine the quotient and remainder.
x² - 4x____________
x + 10 | x³ + 6x² - 34x + 61
x³ + 10x²
-----------
-4x² - 34x
-4x² - 40x
-----------
6x + 61
Step 7: Final Division
We divide the leading term 6x by the leading term of the divisor x, which gives us 6. This is the last term of our quotient, so we write it next to -4x. Then, we multiply 6 by (x + 10) to get 6x + 60, and we write this below 6x + 61. This step completes the main division process. Dividing the remaining terms allows us to determine the final term of the quotient. Multiplying this term by the divisor sets up the final subtraction, which will reveal the remainder.
x² - 4x + 6________
x + 10 | x³ + 6x² - 34x + 61
x³ + 10x²
-----------
-4x² - 34x
-4x² - 40x
-----------
6x + 61
6x + 60
Step 8: Determine the Remainder
Finally, we subtract 6x + 60 from 6x + 61, which gives us 6x + 61 - (6x + 60) = 1. This is our remainder. The remainder is the amount left over after the division process is complete. It represents the portion of the dividend that could not be evenly divided by the divisor. Identifying the remainder is crucial for expressing the final answer in the correct form.
x² - 4x + 6________
x + 10 | x³ + 6x² - 34x + 61
x³ + 10x²
-----------
-4x² - 34x
-4x² - 40x
-----------
6x + 61
6x + 60
------
1
Step 9: Write the Final Answer
Our quotient is x² - 4x + 6, and our remainder is 1. We write the final answer as the quotient plus the remainder divided by the divisor: x² - 4x + 6 + 1/(x + 10). Expressing the final answer correctly involves combining the quotient and the remainder. The remainder is written as a fraction with the divisor as the denominator. This form accurately represents the result of the polynomial division and is essential for various algebraic manipulations.
Therefore, (x³ + 6x² - 34x + 61) ÷ (x + 10) = x² - 4x + 6 + 1/(x + 10).
Identifying the Correct Option
Looking at the options provided:
a. x² - 4x + 8 b. x² - 4x + 8 - 4/(x + 10) c. x² - 4x + 3 + 1/(x + 10) d. x² - 4x + 6 + 1/(x + 10)
The correct answer is d. x² - 4x + 6 + 1/(x + 10), which matches our step-by-step solution.
Common Mistakes to Avoid
Polynomial division can be tricky, so let's talk about some common pitfalls to watch out for:
- Forgetting to align like terms: Always make sure to align terms with the same exponent when subtracting. This keeps your work organized and prevents errors.
- Incorrectly multiplying: Double-check your multiplication steps to ensure you're distributing correctly.
- Sign errors: Pay close attention to signs when subtracting. A small mistake can throw off the entire solution.
- Skipping terms: If a term is missing in the dividend (e.g., no x term), include a placeholder (like 0x) to maintain the proper order.
- Rushing through the steps: Take your time and work methodically through each step. Accuracy is more important than speed.
By being mindful of these common mistakes, you can significantly improve your accuracy and confidence in polynomial division.
Practice Makes Perfect
The best way to master polynomial division is to practice, practice, practice! Try working through different examples with varying degrees of complexity. You can find plenty of practice problems in textbooks, online resources, and worksheets. Start with simpler problems and gradually move on to more challenging ones. The more you practice, the more comfortable and confident you'll become with the process.
Remember, polynomial division is a foundational skill in algebra. Mastering it will not only help you in your current studies but also prepare you for more advanced topics in mathematics and other fields. So, keep practicing, stay focused, and don't be afraid to ask for help when you need it. You've got this!
Conclusion
So there you have it! We've successfully divided (x³ + 6x² - 34x + 61) by (x + 10) using long division. Remember, the key is to break it down step by step, stay organized, and watch out for those common mistakes. Keep practicing, and you'll be a polynomial division pro in no time! Understanding polynomial division is not just about solving equations; it's about developing a deeper understanding of algebraic relationships and problem-solving strategies. Keep up the great work, guys, and happy dividing!