How to Solve or Evaluate the Polynomial Expression Given a Specific Condition

The given problem involves evaluating or solving a polynomial expression under a specific condition. Here, we will explore the polynomial expression x^4 - 2x^3 - 2x 5 given that x frac{1}{x} sqrt{2} 1. The key steps include transforming the given condition and then solving the polynomial under this condition.

Understanding the Given Condition

The condition x frac{1}{x} sqrt{2} 1 can be manipulated to simplify the polynomial expression. Let's start by finding a quadratic form to work with.

Step-by-Step Solution

Step 1: Use the given condition to express a quadratic equation.

We have x frac{1}{x} sqrt{2} 1. Squaring both sides, we obtain:

(x frac{1}{x})^2 (sqrt{2} 1)^2

x^2 2 frac{1}{x^2} 2 2sqrt{2} 1

x^2 frac{1}{x^2} 2sqrt{2} 1

From the given condition, x^2 - x sqrt{2}x - 1, we can write:

x^2 - x^2 - 2x 5 2x^2 - 2x - 2sqrt{2}x 6

Simplifying further, we get:

2(sqrt{2}x - 1) - 2sqrt{2}x 6 4

Thus, the expression evaluates to:

Result: x^4 - 2x^3 - 2x 5 4

Evaluation via Quadratic Equation

Another approach to solving this is through the quadratic formula. Given the equation x frac{1}{x} sqrt{2} 1, we can form a quadratic equation:

x^2 - (sqrt{2} 1)x 1 0

Using the quadratic formula, where a 1, b -(sqrt{2} 1), c 1:

x frac{-b pm sqrt{b^2 - 4ac}}{2a} frac{sqrt{2} 1 pm sqrt{(sqrt{2} 1)^2 - 4}}{2}

x frac{sqrt{2} 1 pm sqrt{3 2sqrt{2} - 4}}{2} frac{sqrt{2} 1 pm sqrt{2sqrt{2} - 1}}{2}

Thus, the roots are:

x_1 frac{sqrt{2} 1 sqrt{2sqrt{2} - 1}}{2}

x_2 frac{sqrt{2} 1 - sqrt{2sqrt{2} - 1}}{2}

Substitute these roots back into the polynomial:

x_1^4 - 2x_1^3x_1^2 - 2x_1 5

x_2^4 - 2x_2^3x_2^2 - 2x_2 5

Both substitutions result in the value 4, as expected from the earlier simplification.

Conclusion

In conclusion, the polynomial expression x^4 - 2x^3 - 2x 5 evaluates to 4 given the condition x frac{1}{x} sqrt{2} 1. This demonstrates the power of algebraic manipulation and quadratic equations in solving polynomial expressions under specific conditions.