Question

What is the solution to x6 – 6x 5 15x 4 – 20x 3 15x 2 – 6x 1 ≥ 0? x = 0 x = 1 all real numbers all real numbers except zero

Answer 1

The solution to the inequality x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x - 1 ≥ 0 is x ≤ -1, x > 1, or -1 < x < 0.

Step-by-step explanation:

The given equation is x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x - 1 ≥ 0. To find its solution, we can analyze the sign of each term and determine the intervals where the inequality is true. Starting from the highest power of x:

1. When x^6 is positive, the inequality is true. This occurs when x < -1 or x > 1.
2. When -6x^5 is negative, the inequality is false. This occurs when -1 < x < 0.
3. When 15x^4 is positive, the inequality is true. This occurs when x < -1 or x > 1.
4. When -20x^3 is negative, the inequality is false. This occurs when -1 < x < 0.
5. When 15x^2 is positive, the inequality is true. This occurs when x < -1 or x > 1.
6. When -6x is negative, the inequality is false. This occurs when -1 < x < 0.
7. The constant term (-1) does not affect the inequality.

Therefore, the solution to the inequality is x ≤ -1, x > 1, or -1 < x < 0.