Solving The Cubic Equation: 3x³ - X² - X - 1 = 0

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Hey guys! Today, we're diving into a fun little math problem: solving the cubic equation 3x³ - x² - x - 1 = 0. Cubic equations might seem intimidating at first, but don't worry, we'll break it down step by step. We'll explore different methods and strategies to tackle this equation, making sure you understand each part of the process. So, grab your thinking caps, and let’s get started!

Understanding Cubic Equations

Before we jump into solving our specific equation, let's quickly recap what cubic equations are all about. A cubic equation is a polynomial equation of degree three. This means the highest power of the variable (in our case, x) is 3. The general form of a cubic equation is:

ax³ + bx² + cx + d = 0

Where a, b, c, and d are constants, and a is not equal to zero (otherwise, it would be a quadratic equation). Cubic equations can have up to three solutions, which can be real or complex numbers. Finding these solutions is what we're aiming for.

Now, let's relate this back to our equation: 3x³ - x² - x - 1 = 0. Here, we have:

  • a = 3
  • b = -1
  • c = -1
  • d = -1

Knowing this helps us understand the structure of the equation and how to approach it. Solving cubic equations can sometimes be tricky, and there isn't a one-size-fits-all method. However, we have several tools at our disposal, which we'll explore in the following sections.

Understanding the nature of cubic equations is crucial for anyone looking to master algebra. The coefficients and constants play a significant role in determining the roots of the equation. For instance, the Rational Root Theorem can be incredibly useful in identifying potential rational solutions, which we will discuss later. Moreover, the shape of a cubic function's graph can provide visual clues about the number and nature of its roots. A cubic function can have one, two, or three real roots, depending on how the curve intersects the x-axis. By grasping these fundamental concepts, you'll be better equipped to tackle not just this equation, but a wide array of polynomial problems. So, let's move on to the first method we can use to try and solve our cubic equation.

Method 1: The Rational Root Theorem

One of the first tools we can use to try and solve cubic equations is the Rational Root Theorem. This theorem helps us find potential rational roots (roots that can be expressed as a fraction) of the equation. It states that if a polynomial equation has integer coefficients, then any rational root of the equation must be of the form:

p/q

Where p is a factor of the constant term (d in our general form) and q is a factor of the leading coefficient (a in our general form).

For our equation, 3x³ - x² - x - 1 = 0:

  • The constant term d is -1, and its factors are ±1.
  • The leading coefficient a is 3, and its factors are ±1, ±3.

Therefore, the possible rational roots are:

±1 / ±1 = ±1

±1 / ±3 = ±1/3

So, the potential rational roots are 1, -1, 1/3, and -1/3. Now, we need to test these values to see if any of them actually satisfy the equation. We can do this by substituting each value for x in the equation and checking if the result is zero. Let’s start with 1:

3(1)³ - (1)² - (1) - 1 = 3 - 1 - 1 - 1 = 0

Great! We found a root. x = 1 is a solution to the equation. This is a significant step because once we find one root, we can reduce the cubic equation to a quadratic equation, which is much easier to solve.

Using the Rational Root Theorem is a systematic way to narrow down the possible solutions of a polynomial equation. It's like having a roadmap that guides you through the potential roots, rather than blindly guessing. The beauty of this theorem lies in its simplicity and effectiveness. By identifying the factors of the constant term and the leading coefficient, we can create a list of candidates that are likely to be the roots. However, it's important to remember that this theorem only gives us potential rational roots; it doesn't guarantee that we'll find one. Sometimes, the roots might be irrational or complex, in which case we would need to use other methods. But for many cubic equations, the Rational Root Theorem is an excellent starting point. Now that we've found one root, let's see how we can use it to simplify the equation further.

Method 2: Synthetic Division or Polynomial Long Division

Now that we've found one root (x = 1) using the Rational Root Theorem, we can use this information to simplify our cubic equation. We'll do this by dividing the original cubic polynomial by (x - 1). This process will reduce the cubic equation to a quadratic equation, which we can then solve using familiar methods like factoring or the quadratic formula. There are two main ways to perform this division: synthetic division and polynomial long division. Let's look at both.

Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear factor. It's generally quicker and easier to use than polynomial long division, especially for simple linear divisors like (x - 1). Here’s how we can use synthetic division for our equation 3x³ - x² - x - 1 = 0 and the root x = 1:

  1. Write down the coefficients of the polynomial: 3, -1, -1, -1.
  2. Write the root (1) to the left.
  3. Bring down the first coefficient (3) to the bottom row.
  4. Multiply the root (1) by the number you just brought down (3) and write the result (3) under the next coefficient (-1).
  5. Add the numbers in that column (-1 + 3 = 2) and write the result (2) in the bottom row.
  6. Repeat steps 4 and 5 for the remaining coefficients.

Here’s how it looks:

1 | 3 -1 -1 -1
  | 3 2 1
  ------------
  3 2 1 0

The numbers in the bottom row (3, 2, 1) are the coefficients of the quotient polynomial, and the last number (0) is the remainder. Since the remainder is 0, this confirms that (x - 1) is indeed a factor of the cubic polynomial. The quotient polynomial is 3x² + 2x + 1.

Polynomial Long Division

If you're not comfortable with synthetic division, or if the divisor is more complex, you can use polynomial long division. This method is similar to long division with numbers. Here’s how it works for our equation:

 3x² + 2x + 1
x - 1 | 3x³ - x² - x - 1
       - (3x³ - 3x²)
       -------------
            2x² - x
            - (2x² - 2x)
            -----------
                 x - 1
                 - (x - 1)
                 ---------
                      0

As you can see, the result is the same: the quotient is 3x² + 2x + 1, and the remainder is 0. Now we have a quadratic equation to solve!

Both synthetic division and polynomial long division are powerful techniques for simplifying polynomial equations. Choosing the right method often depends on the complexity of the divisor and your personal preference. Synthetic division is generally faster and more efficient for linear divisors, while polynomial long division is more versatile and can handle divisors of higher degrees. The key takeaway here is that by using either method, we've successfully reduced our cubic equation to a more manageable quadratic equation. This is a significant step forward, as quadratic equations have well-established methods for finding their solutions. In the next section, we'll apply one of these methods to solve the quadratic equation we've obtained.

Method 3: Solving the Quadratic Equation

After dividing our cubic equation 3x³ - x² - x - 1 = 0 by (x - 1), we obtained the quadratic equation 3x² + 2x + 1 = 0. Now, we need to solve this quadratic equation to find the remaining roots. There are several methods to solve quadratic equations, but the most common ones are factoring, completing the square, and using the quadratic formula. In this case, factoring isn’t straightforward, so let’s use the quadratic formula, which always works.

The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, 3x² + 2x + 1 = 0:

  • a = 3
  • b = 2
  • c = 1

Plugging these values into the quadratic formula, we get:

x = (-2 ± √(2² - 4 * 3 * 1)) / (2 * 3)

x = (-2 ± √(4 - 12)) / 6

x = (-2 ± √(-8)) / 6

Since we have a negative number under the square root, the roots will be complex. We can simplify √(-8) as follows:

√(-8) = √(8 * -1) = √(4 * 2 * -1) = 2i√2

Where i is the imaginary unit (√-1).

So, our solutions are:

x = (-2 ± 2i√2) / 6

We can simplify this by dividing both the real and imaginary parts by 2:

x = (-1 ± i√2) / 3

Thus, the two complex roots are:

x₁ = (-1 + i√2) / 3

x₂ = (-1 - i√2) / 3

Solving the quadratic equation is a fundamental skill in algebra, and the quadratic formula is a powerful tool for this purpose. It allows us to find the roots of any quadratic equation, regardless of whether they are real or complex. The discriminant (b² - 4ac) plays a crucial role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots; if it's zero, the equation has one real root (a repeated root); and if it's negative, as in our case, the equation has two complex conjugate roots. Understanding how to apply the quadratic formula and interpret the discriminant is essential for mastering quadratic equations. Now that we've found the complex roots, we have all the solutions to our original cubic equation.

Final Solutions

Alright, guys! We’ve tackled the cubic equation 3x³ - x² - x - 1 = 0 using a combination of techniques, and now we have all the solutions. Let’s recap:

  1. Using the Rational Root Theorem, we found one real root: x = 1.
  2. We then used synthetic division (or polynomial long division) to divide the cubic polynomial by (x - 1), reducing it to a quadratic equation: 3x² + 2x + 1 = 0.
  3. Finally, we used the quadratic formula to solve the quadratic equation, which gave us two complex roots:
    • x₁ = (-1 + i√2) / 3
    • x₂ = (-1 - i√2) / 3

So, the three solutions to the cubic equation 3x³ - x² - x - 1 = 0 are:

  • x = 1
  • x = (-1 + i√2) / 3
  • x = (-1 - i√2) / 3

These are the complete set of solutions for our cubic equation. This journey demonstrates how multiple algebraic techniques can be combined to solve complex problems. The Rational Root Theorem helps us find potential rational roots, division simplifies the equation, and the quadratic formula provides a definitive way to solve quadratic equations. By mastering these methods, you'll be well-equipped to tackle a wide range of polynomial equations.

Solving cubic equations might seem daunting at first, but by breaking down the problem into smaller, manageable steps, we can find the solutions. Remember, practice makes perfect, so keep working on different equations to hone your skills. You've got this! Keep exploring the fascinating world of mathematics, and there's no limit to what you can achieve. Until next time, keep those brains buzzing!