Synthetic Division: Proving A Root Of A Polynomial

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Hey math enthusiasts! Ever wondered how to crack the code of polynomial equations? Well, today, we're diving deep into synthetic division, a super handy technique that helps us find the roots (or zeros) of a polynomial. And we're going to use it to show that 4 is a solution to the equation: x³ - 2x² - 11x + 12 = 0. Get ready, because it's going to be a fun ride. Synthetic division is a streamlined process for dividing a polynomial by a linear expression of the form (x - k). If the remainder is zero, it means that k is a root of the polynomial. This is because, according to the Factor Theorem, if a polynomial f(x) has a root at x = k, then (x - k) is a factor of f(x). We are going to put this knowledge to use, so you will see just how easy it is, so let's get started. We will start by stating the question and restating the equation, so we have a good grasp of the problem. We will then perform the synthetic division, and from there we will draw our conclusions.

Before we begin, remember that roots are the values of x that make the equation equal to zero. In other words, when you plug in a root into the polynomial, the entire expression evaluates to zero. So, our mission is to show that when x = 4, the equation x³ - 2x² - 11x + 12 = 0 holds true. This is the power of synthetic division—it provides a quick and efficient way to test potential roots. Are you ready to dive in? Let's go!

Understanding the Polynomial Equation and Synthetic Division Setup

Alright, let's break down what we have. Our polynomial equation is x³ - 2x² - 11x + 12 = 0. The degree of the polynomial is 3 (because of the x³ term), which means we're dealing with a cubic equation. Now, we want to check if x = 4 is a solution. Here's how we set up the synthetic division:

  1. Write down the coefficients: The coefficients of our polynomial are 1 (from x³), -2 (from -2x²), -11 (from -11x), and 12 (the constant term). We'll write these down in a row: 1, -2, -11, 12.
  2. Place the potential root: Since we're testing if 4 is a root, we place 4 to the left of these coefficients. It looks like this: 4 | 1 -2 -11 12.

That's it, that's the starting line for synthetic division. This setup is crucial, so make sure you've got it right. The process itself is surprisingly straightforward. The goal is to see if we get a remainder of 0. If we do, we've confirmed that 4 is indeed a root. Let's start the steps to solve this. The beauty of synthetic division lies in its simplicity. Let's go through the steps of this process.

Performing Synthetic Division: The Calculation Steps

Now, let's get into the heart of synthetic division. Follow these steps to complete the process:

  1. Bring down the first coefficient: Bring down the first coefficient (which is 1) below the line: 4 | 1 -2 -11 12 1

  2. Multiply and add: Multiply the number you just brought down (1) by the potential root (4), and write the result (4) under the next coefficient (-2). Then, add the numbers in that column (-2 + 4 = 2): 4 | 1 -2 -11 12 | 4

    1 2

  3. Repeat the process: Multiply the result (2) by the potential root (4), which gives you 8. Write 8 under the next coefficient (-11) and add: -11 + 8 = -3. 4 | 1 -2 -11 12 | 4 8

    1 2 -3

  4. Final step: Multiply -3 by 4, giving you -12. Write -12 under the last term (12) and add: 12 + (-12) = 0. 4 | 1 -2 -11 12 | 4 8 -12

    1 2 -3 0

And there you have it! The last number in the bottom row is the remainder. In this case, the remainder is 0. This is the key piece of information. The remainder being zero tells us that the potential root, 4, is indeed a root of the polynomial. This means that if you plug in 4 into the original equation, you will get zero. So, we've successfully used synthetic division to verify that 4 is a solution.

Interpreting the Results and Conclusion

Since the remainder is 0, we've proven that x = 4 is a root of the equation x³ - 2x² - 11x + 12 = 0. What does this mean in practical terms? Well, it means that (x - 4) is a factor of the polynomial. The other numbers in the bottom row (1, 2, -3) represent the coefficients of the quotient, which is a quadratic equation: x² + 2x - 3. This is what you get when you divide the original polynomial by (x - 4). So, you have transformed a cubic equation into a simpler quadratic equation. You could even solve this quadratic equation to find the other roots of the original cubic equation. We are not going to solve for the other roots in this example, but it shows how powerful synthetic division is.

In summary:

  • We set up the synthetic division with the coefficients of the polynomial and the potential root (4).
  • We performed the multiplication and addition steps.
  • We got a remainder of 0.
  • This confirms that 4 is a root of the polynomial equation.

Therefore, we have successfully used synthetic division to verify that 4 is a solution to the polynomial equation x³ - 2x² - 11x + 12 = 0. Pretty cool, right? This is the power of synthetic division - it provides a clear and concise way to test and confirm the roots of a polynomial. The technique is very easy to use, and it opens up a new world to understanding and solving polynomial equations. Are you ready for more? The method is very common in mathematics and is a staple in math courses.

Further Exploration: What's Next?

Now that you've seen how to use synthetic division to prove a root, what's next? Here are a few ideas to build on your new skills:

  • Practice, practice, practice: Try solving different polynomial equations. The more you work with synthetic division, the more comfortable you'll become. Experiment with different potential roots and polynomials of different degrees.
  • Explore the Factor Theorem: Understand how the Factor Theorem connects roots and factors. This theorem is the foundation of why synthetic division works.
  • Find all the roots: Once you've found one root using synthetic division, use the resulting quotient (the other polynomial) to find the remaining roots. You can use factoring, the quadratic formula, or more synthetic division (if applicable).
  • Graphing calculators: Use graphing calculators to see the graphical representation of the polynomial equations and where the roots lie on the graph. This visual confirmation can deepen your understanding.

By continuing to practice and explore, you'll gain even more confidence in your ability to tackle polynomial equations. Keep up the great work! You've successfully navigated the world of synthetic division and its applications. Keep exploring, keep learning, and don't be afraid to challenge yourself with more complex problems. The more you engage with the material, the better you'll become at solving polynomial equations. Enjoy the journey!