Synthetic Division Explained Is 5 A Zero Of F(x) = X² - 7x + 10

Hey guys! Let's dive into a crucial concept in polynomial functions: synthetic division and how it helps us determine if a given number is a zero of a polynomial. In this article, we'll tackle the specific function f(x) = x² - 7x + 10 and the potential zero k = 5. We'll walk through the process step-by-step, making sure you understand not just the how, but also the why behind it all. So, buckle up, and let's get started!

What are Polynomial Zeros?

Before we jump into the nitty-gritty, let's quickly recap what a polynomial zero actually is. Simply put, a zero of a polynomial function f(x) is a value of x that makes the function equal to zero. In other words, if f(k) = 0, then k is a zero of the polynomial. These zeros are incredibly important because they tell us where the graph of the polynomial intersects the x-axis. They are also known as roots or solutions of the polynomial equation f(x) = 0.

Finding these zeros is a fundamental problem in algebra, and there are several techniques we can use. One such technique, and the one we'll be focusing on today, is synthetic division. Synthetic division provides a streamlined way to divide a polynomial by a linear factor of the form (x - k). This is particularly useful because the Remainder Theorem tells us that the remainder we get from this division is equal to f(k). So, if the remainder is zero, we know that k is a zero of the polynomial! Understanding zeros is crucial for analyzing polynomial behavior and solving related equations. They provide key insights into the graph and algebraic properties of the polynomial function.

Why do we care about zeros? Well, they're not just abstract mathematical concepts. Zeros have real-world applications in various fields, including engineering, physics, and economics. For instance, in engineering, finding the zeros of a polynomial can help determine the stability of a system. In physics, they can represent equilibrium points. And in economics, they might represent break-even points in a cost-benefit analysis. So, grasping how to find zeros, especially with tools like synthetic division, is a valuable skill to have. The ability to efficiently find zeros can significantly simplify complex problems and provide valuable insights into the underlying relationships represented by the polynomial function. Zeros act as critical points that define the function's behavior and its interactions with other mathematical entities.

Synthetic Division: A Step-by-Step Guide

Now, let's get to the heart of the matter: synthetic division. Imagine synthetic division as a shortcut, a more efficient way to perform polynomial division when you're dividing by a linear factor (x - k). Instead of writing out all the terms and doing long division, we focus on the coefficients and use a streamlined process. It might seem a little weird at first, but trust me, with a bit of practice, you'll be a synthetic division pro in no time!

Let's break down the steps involved in synthetic division. First, write down the coefficients of the polynomial in a row, making sure to include zeros for any missing terms. For example, if we had x³ + 2x - 1, we would write down 1, 0, 2, -1 (notice the zero for the missing x² term). Next, write the value of k (the potential zero) to the left. Draw a horizontal line below the coefficients. Now, the fun begins! Bring down the first coefficient below the line. Multiply this number by k and write the result under the next coefficient. Add the two numbers in that column and write the sum below the line. Repeat this process of multiplying by k and adding until you've reached the last coefficient. The last number you get below the line is the remainder. Synthetic division simplifies the process of polynomial division by focusing on the essential numerical relationships between coefficients and the potential zero.

The numbers below the line (excluding the remainder) are the coefficients of the quotient polynomial, which is one degree lower than the original polynomial. The remainder is crucial because, as we mentioned earlier, it tells us the value of f(k). If the remainder is zero, then k is a zero of the polynomial. If the remainder is not zero, then k is not a zero, and the remainder is the value of f(k). Let's say we're dividing a cubic polynomial (degree 3) by (x - k). The quotient will be a quadratic polynomial (degree 2). The coefficients we get from synthetic division directly correspond to the coefficients of this quadratic. The constant term of the quotient, along with the remainder, provides valuable information about the polynomial's behavior near x = k. Synthetic division not only helps in identifying zeros but also provides a concise representation of the division process, making it a powerful tool in polynomial analysis. Mastering synthetic division empowers you to quickly determine polynomial zeros and understand the structure of polynomial division.

Applying Synthetic Division to f(x) = x² - 7x + 10 and k = 5

Okay, let's get our hands dirty and apply synthetic division to our specific problem: f(x) = x² - 7x + 10 and k = 5. This is where the rubber meets the road, guys! We'll walk through each step, so you can see exactly how it works in practice. This example will solidify your understanding and give you the confidence to tackle similar problems on your own.

First, we identify the coefficients of the polynomial. In this case, we have 1 (for x²), -7 (for -7x), and 10 (the constant term). We write these coefficients in a row: 1 -7 10. Next, we write the potential zero, k = 5, to the left. Now, we draw a horizontal line below the coefficients. We're all set to start the synthetic division process! The key here is to keep the numbers organized and follow the steps carefully. A slight mistake can throw off the whole calculation, so double-checking your work is always a good idea. Applying synthetic division systematically ensures accurate results and a clear understanding of the polynomial's behavior.

Now, let's execute the synthetic division steps. Bring down the first coefficient, 1, below the line. Multiply this 1 by 5 (our k value) and write the result, 5, under the next coefficient, -7. Add -7 and 5 to get -2, and write this below the line. Multiply -2 by 5 and write the result, -10, under the last coefficient, 10. Finally, add 10 and -10 to get 0, and write this below the line. This 0 is our remainder! Remember, the remainder is crucial because it tells us the value of f(k). Since our remainder is 0, we know that f(5) = 0, meaning that 5 is indeed a zero of the polynomial. Interpreting the remainder is the final piece of the puzzle in synthetic division, providing the answer to whether the tested value is a zero of the polynomial.

Is 5 a Zero? The Verdict and the Value of f(5)

So, what's the verdict? Is 5 a zero of the polynomial f(x) = x² - 7x + 10? Drumroll, please… Yes! Based on our synthetic division calculation, the remainder is 0. As we discussed earlier, this means that f(5) = 0, confirming that 5 is indeed a zero of the function. This is a fantastic result! It demonstrates the power and efficiency of synthetic division in identifying polynomial zeros. Understanding how to interpret the results of synthetic division is key to solving polynomial equations and analyzing polynomial behavior.

But we've also answered another part of the question: what is the value of f(5)? We already know the answer, but it's worth reiterating. Because the remainder from the synthetic division is 0, f(5) = 0. This reinforces the connection between zeros and the function's value at those points. The fact that f(5) = 0 also means that the graph of f(x) intersects the x-axis at x = 5. This visual representation of the zero can be incredibly helpful in understanding the overall shape and behavior of the polynomial function. The zero, the function's value, and the graph's intersection are all interconnected concepts that synthetic division helps us understand.

Let's recap the coefficients of the quotient polynomial as well. From our synthetic division, the numbers below the line (excluding the remainder) are 1 and -2. This tells us that the quotient polynomial is x - 2. In other words, f(x) can be factored as (x - 5)(x - 2). This factorization further confirms that 5 is a zero, as well as revealing that 2 is another zero of the polynomial. This connection between synthetic division, zeros, and factorization highlights the versatility of this technique. Synthetic division not only identifies zeros but also provides insights into the polynomial's factors and its overall algebraic structure.

Conclusion: Mastering Synthetic Division for Polynomial Zeros

Alright, guys, we've reached the end of our journey into synthetic division and its application in finding polynomial zeros. We started by understanding what polynomial zeros are and why they're important. Then, we dove into the step-by-step process of synthetic division, learning how to set it up and execute the calculations. Finally, we applied this knowledge to the specific function f(x) = x² - 7x + 10 and the potential zero k = 5, successfully determining that 5 is indeed a zero.

Hopefully, you now have a solid grasp of how to use synthetic division to identify zeros of polynomial functions. Remember, practice makes perfect! The more you work through examples, the more comfortable and confident you'll become with this technique. Don't be afraid to make mistakes – they're part of the learning process. The key is to understand the steps, the logic behind them, and how to interpret the results. Consistent practice with synthetic division will transform it from a challenging concept into a valuable tool in your mathematical arsenal.

Synthetic division is more than just a computational trick; it's a powerful tool for understanding the structure and behavior of polynomials. It connects the dots between zeros, factors, and the function's value. By mastering this technique, you'll be well-equipped to tackle a wide range of polynomial problems. So, keep practicing, keep exploring, and keep those polynomial zeros in sight! Embrace the power of synthetic division to unlock deeper insights into the world of polynomial functions.