Understanding the Problem: Solving the Equation and Evaluating the Expression

The problem asks us to find the value of a specific expression involving powers of \(x\), given a quadratic equation that \(x\) satisfies. The given equation is \(x^2 + x + 1 = 0\), and the expression is \(x^{199} + x^{200} + x^{201}\).

Analyzing the Special Quadratic Equation \(x^2 + x + 1 = 0\)

The equation \(x^2 + x + 1 = 0\) is a very special quadratic equation. Its roots are related to the complex cube roots of unity. Let's call the roots of this equation \(\omega\) and \(\omega^2\). These roots have the following properties:

• They satisfy the equation: \(\omega^2 + \omega + 1 = 0\) and \((\omega^2)^2 + \omega^2 + 1 = 0\), which simplifies to \(\omega^4 + \omega^2 + 1 = 0\). Since \(\omega^3 = 1\), this is \(\omega + \omega^2 + 1 = 0\).
• Multiplying the equation \(x^2 + x + 1 = 0\) by \((x-1)\) gives \((x-1)(x^2 + x + 1) = 0\), which is \(x^3 - 1 = 0\). This means \(x^3 = 1\). So, any value of \(x\) satisfying \(x^2 + x + 1 = 0\) must also satisfy \(x^3 = 1\).
• The sum of the roots is \(-1\) and the product of the roots is \(1\), which is consistent with properties of roots of unity.

The key property we will use is that if \(x\) is a root of \(x^2 + x + 1 = 0\), then \(x^3 = 1\). This property allows us to simplify higher powers of \(x\).

Simplifying Powers of x Using \(x^3 = 1\)

Since \(x^3 = 1\), any power of \(x\) can be simplified by finding the remainder of the exponent when divided by 3. This is because \(x^n = x^{3q + r} = (x^3)^q \cdot x^r = 1^q \cdot x^r = x^r\), where \(r\) is the remainder when \(n\) is divided by 3.

Let's simplify the exponents in the expression \(x^{199} + x^{200} + x^{201}\):

• For \(x^{199}\): Divide 199 by 3. \(199 = 3 \times 66 + 1\). The remainder is 1. So, \(x^{199} = x^1 = x\).
• For \(x^{200}\): Divide 200 by 3. \(200 = 3 \times 66 + 2\). The remainder is 2. So, \(x^{200} = x^2\).
• For \(x^{201}\): Divide 201 by 3. \(201 = 3 \times 67 + 0\). The remainder is 0. So, \(x^{201} = x^0 = 1\).

Evaluating the Expression \(x^{199} + x^{200} + x^{201}\)

Now substitute the simplified terms back into the expression:

\(x^{199} + x^{200} + x^{201} = x + x^2 + 1\)

We know from the original equation that \(x^2 + x + 1 = 0\).

Therefore, the value of the expression is 0.

\(x^{199} + x^{200} + x^{201} = x + x^2 + 1 = 0\)

Summary of the Solution Steps

Here's a quick recap of the steps taken to solve this problem:

1. Recognize that the equation \(x^2 + x + 1 = 0\) implies \(x^3 = 1\) for its roots.
2. Use the property \(x^3 = 1\) to simplify the higher powers of \(x\) in the expression by finding the exponent modulo 3.
3. Simplify \(x^{199}\) to \(x\), \(x^{200}\) to \(x^2\), and \(x^{201}\) to \(1\).
4. Substitute these simplified terms back into the expression \(x^{199} + x^{200} + x^{201}\) to get \(x + x^2 + 1\).
5. Use the given equation \(x^2 + x + 1 = 0\) to find the final value of the simplified expression.

Revision Table: Key Concepts in Solving \(x^2 + x + 1 = 0\) Problems

Additional Information: Cube Roots of Unity

The cube roots of unity are the solutions to the equation \(z^3 = 1\). These solutions are:

• \(z=1\) (the real root)
• \(z = \omega = e^{i 2\pi/3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\)
• \(z = \omega^2 = e^{i 4\pi/3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}\)

The equation \(x^2 + x + 1 = 0\) is obtained by factoring \(x^3 - 1 = 0\) as \((x-1)(x^2 + x + 1) = 0\). The roots of \(x-1=0\) is \(x=1\). The roots of \(x^2 + x + 1 = 0\) are the other two roots, \(\omega\) and \(\omega^2\).

Properties of \(\omega\) and \(\omega^2\):

• \(1 + \omega + \omega^2 = 0\) (as seen in the problem)
• \(\omega^3 = 1\)
• \(\omega^2\) is the conjugate of \(\omega\).
• Any positive integer power of \(\omega\) can be reduced to \(\omega\), \(\omega^2\), or \(1\) based on the exponent modulo 3.

Understanding these properties is fundamental for solving problems involving the roots of \(x^2 + x + 1 = 0\).