Solving The Cubic Equation: X³ - 2x² - 9x + 18 = 0

Hey math enthusiasts! Today, we're diving into the world of cubic equations and tackling the problem: x³ - 2x² - 9x + 18 = 0. Don't worry, it might seem a bit intimidating at first, but trust me, we'll break it down step-by-step to make it super clear and easy to understand. Cubic equations are those with a variable raised to the power of 3, and solving them involves finding the values of 'x' that make the equation true. Let's get started with this awesome math journey. In this article, we'll explore different methods to solve this equation, including factoring, and the rational root theorem. This will help you find the roots (solutions) of the equation. So, grab your pencils, and let's unravel this mathematical puzzle together. We'll start with the basics and gradually move towards more advanced techniques, ensuring everyone can follow along. Remember, the goal is not just to find the answer but to understand the 'why' behind each step. Alright, buckle up, and let's do this!

Factoring by Grouping: A Clever Approach

Alright, guys, let's kick things off with a cool technique called factoring by grouping. It's like a secret weapon for solving cubic equations when they're in a specific form, and guess what? Our equation, x³ - 2x² - 9x + 18 = 0, is perfect for this! The main idea behind factoring by grouping is to rearrange the terms of the equation and then group them in such a way that we can factor out common terms. This often simplifies the equation and makes it easier to find the solutions. Here's how we're going to break it down. First, we look at the first two terms: x³ - 2x². Notice that both terms have an x² in common. We factor out the x², leaving us with x²(x - 2). Next, let's look at the last two terms: -9x + 18. Here, we can factor out -9, resulting in -9(x - 2). Now, our equation looks like this: x²(x - 2) - 9(x - 2) = 0. See that (x - 2) showing up in both parts? That's a sign we're on the right track! We can factor out the common term (x - 2), and we're left with (x - 2)(x² - 9) = 0. Awesome, right? Factoring by grouping has turned a seemingly complex cubic equation into something much more manageable. What we have now is a product of two factors that equals zero, and we know that if a product is zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for x. This method is incredibly helpful and straightforward to solve our cubic equation. So, we'll take a deep breath and start the next section to continue solving it.

Finding the Roots

Now that we've used factoring by grouping, we're in a great position to find the roots of the equation. We have (x - 2)(x² - 9) = 0. As we said before, if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x. Firstly, let's solve for the factor (x - 2) = 0. Adding 2 to both sides, we get x = 2. Great! We've found our first root. Secondly, we'll deal with the factor (x² - 9) = 0. This is a difference of squares, and it can be easily factored into (x - 3)(x + 3) = 0. Setting each of these factors equal to zero, we get x - 3 = 0, which gives us x = 3, and x + 3 = 0, which gives us x = -3. There we have it! We've found all three roots of the cubic equation: x = 2, x = 3, and x = -3. Finding the roots is the ultimate goal when solving an equation, and it allows us to know the points where the function crosses the x-axis in the coordinate plane. Remember, these roots are the values of x that make the original equation true. These roots are like the hidden keys that unlock the solution to our equation. This process shows how a cubic equation can be broken down into simpler factors and then solved systematically to find all the solutions. So now, you can go ahead and try other cubic equations to get the hang of it!

Using the Rational Root Theorem: Another Way

Alright, folks, let's switch gears and explore another powerful method called the Rational Root Theorem. This theorem gives us a systematic way to identify potential rational roots of a polynomial equation, like our cubic equation. The Rational Root Theorem is particularly useful when factoring by grouping doesn't work immediately or when dealing with more complex cubic equations. Here's how it works. First, we need to understand that the Rational Root Theorem states that if a polynomial equation has integer coefficients, then any rational root must be in the form p/q, where p is a factor of the constant term (the term without any x) and q is a factor of the leading coefficient (the coefficient of the highest power of x). In our equation, x³ - 2x² - 9x + 18 = 0, the constant term is 18, and the leading coefficient is 1. The factors of 18 are ±1, ±2, ±3, ±6, ±9, and ±18. The factors of 1 are ±1. Therefore, the possible rational roots (p/q) are ±1, ±2, ±3, ±6, ±9, and ±18. Now we will test each of these potential roots by substituting them into the equation to see if they make the equation equal to zero. Let's start with x = 1: (1)³ - 2(1)² - 9(1) + 18 = 1 - 2 - 9 + 18 = 8 ≠ 0. So, x = 1 is not a root. Let's try x = -1: (-1)³ - 2(-1)² - 9(-1) + 18 = -1 - 2 + 9 + 18 = 24 ≠ 0. Nope. Let's move to x = 2: (2)³ - 2(2)² - 9(2) + 18 = 8 - 8 - 18 + 18 = 0. Hey, x = 2 works! We found our first root. This means that (x - 2) is a factor of the polynomial. With this theorem, we are able to easily find the roots of the cubic equation.

Synthetic Division

Since we've found that x = 2 is a root, we can now use synthetic division to divide the polynomial by (x - 2). Synthetic division is a simplified way to divide a polynomial by a linear factor like (x - 2). It's a quick and efficient method that helps us find the remaining quadratic factor. Here’s how we do it: Write down the coefficients of the polynomial: 1, -2, -9, 18. Write the root (2) to the left of these coefficients. Bring down the first coefficient (1). Multiply the root (2) by this number (1) and write the result (2) under the second coefficient (-2). Add the second column: -2 + 2 = 0. Multiply the root (2) by this result (0) and write the result (0) under the third coefficient (-9). Add the third column: -9 + 0 = -9. Multiply the root (2) by this result (-9) and write the result (-18) under the fourth coefficient (18). Add the fourth column: 18 - 18 = 0. The last number (0) is the remainder, which confirms that (x - 2) is indeed a factor. The other numbers (1, 0, -9) are the coefficients of the resulting quadratic factor. So, after synthetic division, we are left with x² - 9 = 0. That's exactly what we got when we factored by grouping earlier! This gives us the equation (x - 2)(x² - 9) = 0. We've already solved this, finding the roots x = 2, x = 3, and x = -3. The Synthetic Division confirms our finding of the Rational Root Theorem. So we have come up with another great method for solving this cubic equation.

Conclusion: Mastering Cubic Equations

Great job, everyone! We've successfully navigated the world of cubic equations and found the roots of x³ - 2x² - 9x + 18 = 0 using two powerful methods: factoring by grouping and the Rational Root Theorem with synthetic division. We've seen how to break down complex equations into simpler components, making them easier to solve. The roots we found were x = 2, x = 3, and x = -3. Remember that solving these equations is not just about getting the answer; it's about understanding the underlying concepts and developing problem-solving skills. Whether it's factoring by grouping or applying the Rational Root Theorem, each method offers a unique approach to tackling cubic equations. These skills are extremely valuable in various areas of mathematics and beyond. As you continue to practice, you'll become more confident and proficient in solving these types of equations. Keep exploring, keep learning, and keep enjoying the fascinating world of mathematics! With the knowledge gained, you are now well-equipped to tackle similar cubic equations with confidence. Keep up the excellent work, and always remember to embrace the challenge and enjoy the learning process. Keep practicing! And congratulations on solving the cubic equation. You are ready to go, and you can solve many more equations.