Is (x-8) A Factor Of X³ - 3x² - 31x - 72? A Step-by-Step Guide

Hey guys! Ever wondered how to figure out if a polynomial has a certain factor? Today, we're diving deep into a fascinating problem: determining if (x-8) is a factor of the polynomial x³ - 3x² - 31x - 72. This is a classic algebra question that might seem intimidating at first, but trust me, we'll break it down step by step. We'll explore the Factor Theorem, which is our main tool here, and look at how synthetic division can make our lives easier. Plus, we'll chat about common mistakes and how to avoid them. So, grab your thinking caps, and let's get started!

Understanding the Factor Theorem

Let's start with the most important concept for tackling this problem: the Factor Theorem. The Factor Theorem is a cornerstone of polynomial algebra, providing a direct link between the roots of a polynomial and its factors. In simple terms, it states that for a polynomial f(x), if f(c) = 0 for some number c, then (x - c) is a factor of f(x). Conversely, if (x - c) is a factor of f(x), then f(c) = 0. This might sound a bit abstract right now, but let's connect it to our specific problem to make it clearer. In our case, we want to know if (x - 8) is a factor of x³ - 3x² - 31x - 72. According to the Factor Theorem, this is true if and only if plugging in x = 8 into the polynomial results in zero. So, the first thing we need to do is evaluate the polynomial at x = 8. This means we'll substitute 8 for every x in the expression and see what we get. This is a direct application of the theorem and the foundation of our solution. Understanding this connection is crucial because it transforms the problem from a question about factoring polynomials to a straightforward arithmetic calculation. We're essentially checking if 8 is a root of the polynomial. If it is, then (x - 8) is indeed a factor. If not, then it isn't. This simple yet powerful idea is what makes the Factor Theorem such a valuable tool in algebra.

Applying the Factor Theorem to Our Polynomial

Now, let's get our hands dirty and actually apply the Factor Theorem to our specific polynomial, f(x) = x³ - 3x² - 31x - 72. As we discussed, we need to evaluate f(8). This means substituting x = 8 into the polynomial: f(8) = (8)³ - 3(8)² - 31(8) - 72. Okay, let's break this down step by step. First, we calculate 8 cubed, which is 8 * 8 * 8 = 512. Next, we calculate 8 squared, which is 8 * 8 = 64, and then multiply that by 3, giving us 3 * 64 = 192. Then, we multiply 31 by 8, which equals 248. So, now our expression looks like this: f(8) = 512 - 192 - 248 - 72. Now, it's just a matter of doing the subtraction. 512 - 192 = 320. Then, 320 - 248 = 72. Finally, 72 - 72 = 0. So, we've found that f(8) = 0. This is a crucial result! According to the Factor Theorem, since f(8) = 0, this means that (x - 8) is indeed a factor of the polynomial x³ - 3x² - 31x - 72. This confirms our initial approach and demonstrates the power of the Factor Theorem in action. By simply plugging in the value x = 8 and evaluating the polynomial, we were able to definitively determine whether (x - 8) is a factor. This is much simpler than trying to factor the cubic polynomial directly!

Synthetic Division: A Quick Check

While the Factor Theorem gives us a definitive answer, there's another method we can use to check our result and gain even more insight into the polynomial: synthetic division. Synthetic division is a streamlined way to divide a polynomial by a linear factor of the form (x - c). It's often quicker and less prone to errors than long division, especially for higher-degree polynomials. In our case, we'll use synthetic division to divide x³ - 3x² - 31x - 72 by (x - 8). The setup for synthetic division involves writing the coefficients of the polynomial (1, -3, -31, -72) and the value of c (which is 8 in our case) in a specific arrangement. Then, we follow a series of steps: bring down the first coefficient, multiply it by c, add the result to the next coefficient, and repeat. The last number we get is the remainder. If the remainder is 0, it confirms that (x - 8) is a factor, and the other numbers give us the coefficients of the quotient polynomial. Let's walk through the process: We start by writing down the coefficients: 1, -3, -31, and -72. We also write down 8 (from x - 8) to the left. Bring down the 1. Multiply 1 by 8, which gives 8. Add 8 to -3, which gives 5. Multiply 5 by 8, which gives 40. Add 40 to -31, which gives 9. Multiply 9 by 8, which gives 72. Add 72 to -72, which gives 0. The remainder is 0! This confirms what we found using the Factor Theorem. The numbers 1, 5, and 9 are the coefficients of the quotient polynomial, which is x² + 5x + 9. This means that x³ - 3x² - 31x - 72 can be written as (x - 8)(x² + 5x + 9). Synthetic division not only confirms that (x - 8) is a factor but also gives us the other factor, which is a quadratic polynomial. This is a valuable piece of information that can help us further analyze the polynomial, such as finding its other roots.

Common Mistakes to Avoid

When working with the Factor Theorem and synthetic division, there are a few common pitfalls that students often encounter. Let's discuss these so you can avoid them! One of the most frequent mistakes is incorrectly substituting the value of x. Remember, the Factor Theorem states that if (x - c) is a factor, we need to evaluate the polynomial at x = c, not x = -c. So, in our problem, since we're checking if (x - 8) is a factor, we need to substitute x = 8, not x = -8. Substituting the wrong value will lead to an incorrect conclusion. Another common mistake occurs during synthetic division. It's crucial to remember to bring down the first coefficient correctly and to perform the multiplication and addition steps accurately. A small error in any of these steps can throw off the entire calculation and lead to a wrong remainder. Pay close attention to the signs when adding the numbers. For example, if you're adding a negative number, make sure you subtract it correctly. Another mistake is misinterpreting the result of the synthetic division. The remainder is the key indicator of whether the divisor is a factor. If the remainder is 0, then the divisor is a factor. But if the remainder is anything other than 0, then the divisor is not a factor. Also, remember that the numbers you get (other than the remainder) are the coefficients of the quotient polynomial, starting with one degree less than the original polynomial. Finally, it's important to double-check your work, especially if you're unsure about your answer. You can use the Factor Theorem as a quick check of your synthetic division, or vice versa. By being aware of these common mistakes and taking steps to avoid them, you can increase your accuracy and confidence when working with polynomials and the Factor Theorem.

Conclusion: (x-8) is a Factor!

So, let's recap what we've learned and answer the original question: Is (x-8) a factor of x³ - 3x² - 31x - 72? The answer is a resounding yes! We used the Factor Theorem to determine this by evaluating the polynomial at x = 8 and finding that f(8) = 0. This directly implies that (x - 8) is a factor. To further confirm our result and gain additional insight, we also employed synthetic division. Synthetic division not only verified that (x - 8) is a factor (since the remainder was 0) but also provided us with the quotient polynomial, x² + 5x + 9. This means we can express the original polynomial as (x - 8)(x² + 5x + 9). We also discussed common mistakes to avoid, such as incorrect substitution in the Factor Theorem and errors in synthetic division calculations. By understanding these potential pitfalls, you can improve your accuracy and problem-solving skills. The Factor Theorem and synthetic division are powerful tools in algebra, allowing us to efficiently determine factors of polynomials and gain a deeper understanding of their structure. Mastering these techniques will not only help you solve problems like this one but also lay a strong foundation for more advanced concepts in mathematics. So, keep practicing, and you'll become a polynomial pro in no time! Remember, guys, math can be fun, especially when you break it down step by step and understand the underlying principles. Keep exploring, keep learning, and keep challenging yourselves!