Solving the Algebraic Expression Problem

The problem asks us to find the value of a specific algebraic expression, \(x^{18} + x^{12} + x^6 + 1\), given the condition \(x + \frac{1}{x} = \sqrt{3}\). This type of problem often involves finding a specific property of \(x\) from the given condition and then using it to simplify the more complex expression. The condition \(x + \frac{1}{x} = \sqrt{3}\) is a key indicator in this algebraic problem.

Finding a Useful Property of x from the Given Condition

We are given the condition: \[x + \frac{1}{x} = \sqrt{3}\] Let's find a useful property of \(x\) from this equation. A common step with expressions like \(x + \frac{1}{x}\) is to cube it, as \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \). Cubing both sides of the given equation:

Multiplying the last equation by \(x^3\) (assuming \(x \neq 0\), which must be true if \(x + \frac{1}{x} = \sqrt{3}\)):

This is a very important result. The condition \(x + \frac{1}{x} = \sqrt{3}\) implies that \(x^6 = -1\). This property is crucial for simplifying the given algebraic expression.

Simplifying the Given Algebraic Expression using \(x^6 = -1\)

Now we need to find the value of the expression \(x^{18} + x^{12} + x^6 + 1\). We can rewrite the terms with powers of \(x\) as powers of \(x^6\), since we know the value of \(x^6\).

• \(x^{18} = (x^6)^3\)
• \(x^{12} = (x^6)^2\)
• \(x^6\) remains as is

Substitute \(x^6 = -1\) into these terms:

• \(x^{18} = (-1)^3 = -1\)
• \(x^{12} = (-1)^2 = 1\)
• \(x^6 = -1\)

Now substitute these values back into the original algebraic expression:

Let's calculate the sum:

The value of the algebraic expression \(x^{18} + x^{12} + x^6 + 1\) is 0.

Verification and Summary of the Algebraic Solution

We started with the condition \(x + \frac{1}{x} = \sqrt{3}\). We derived that this condition leads to \(x^6 = -1\). Then, we used this result to simplify the target polynomial expression \(x^{18} + x^{12} + x^6 + 1\) by expressing the powers of \(x\) in terms of \(x^6\). \(x^{18} = (x^6)^3 = (-1)^3 = -1\) \(x^{12} = (x^6)^2 = (-1)^2 = 1\) \(x^6 = -1\) The expression becomes \((-1) + (1) + (-1) + 1 = 0\). This confirms our simplification of the algebraic expression.

Thus, the value of \(x^{18} + x^{12} + x^6 + 1\) when \(x + \frac{1}{x} = \sqrt{3}\) is 0. This is a common result encountered in problems involving \(x + \frac{1}{x} = \sqrt{3}\).