Determining the Range of k for Distinct Real Roots of a Polynomial Equation

In mathematics, polynomial equations can often be transformed into simpler forms to facilitate analysis. This article focuses on determining the range of values for k such that the equation x4 - (k-1)x2 - 2 - k 0 has 4 distinct real roots. We will follow a step-by-step approach to solve this problem, ensuring that our analysis aligns with Google's SEO standards.

Transforming the Polynomial Equation

Let's start by transforming the given polynomial equation into a more manageable form using a substitution. Let y x2. This substitution changes the original equation into a quadratic form:

y2 - (k-1)y - 2 - k 0

Critical Steps of the Analysis

Condition 1: Discriminant for Distinct Real Roots

To have two distinct positive roots in the transformed quadratic equation, the discriminant D must be positive. The discriminant is calculated as follows:

D (k-1)2 - 4(2 - k) (k-1)2 - 8 4k k2 - 2k - 7

Therefore, we require:

k2 - 2k - 7 > 0

Step 1: Solving the Discriminant Inequality

The inequality k2 - 2k - 7 > 0 can be solved by finding the roots of the corresponding quadratic equation k2 - 2k - 7 0. Using the quadratic formula:

k (-b ± sqrt{b2 - 4ac}) / (2a) (2 ± sqrt{22 - 4 * 1 * -7}) / 2 (2 ± sqrt{4 28}) / 2 (2 ± 2√2) / 2 1 ± √2

Hence, the roots are:

k_1 1 - √2

k_2 1 √2

Step 2: Testing Intervals Based on Roots

Testing the intervals determined by the roots:

For k > 1 √2: Choose k 5 For (1 - √2) Choose k 0 For k Choose k -5

The combined results of these tests indicate that:

k^2 - 2k - 7 > 0 is satisfied for k > 1 √2 or k .

Condition 2: Positive Roots for y

The roots of the quadratic equation y^2 - (k-1)y - 2 - k 0 are given by:

y1,2 (k-1 ± sqrt{(k-1)2 - 4(2 - k)}) / 2

For both roots y1 and y2 to be positive, the sum and product of the roots must be positive:

y1 y2 k - 1 > 0, which implies k > 1 y1 y2 2 - k > 0, which implies k

Combining these conditions, we have:

1

Conclusion

The final range of values for k such that the equation x4 - (k-1)x2 - 2 - k 0 has 4 distinct real roots is:

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This conclusion reflects a sophisticated analysis of the polynomial equation and ensures that both the theoretical and computational aspects are properly addressed.