Now, we want to find the values of x where this expression is less than or equal to zero. To do this, we need to find the roots of this expression by setting it equal to zero:
х⁶ - 3х⁵ + 8х⁴ - 24х³ - 36х² = 0
Factor out an x²:
x²(х⁴ - 3х³ + 8х² - 24х - 36) = 0
Now, we need to find the roots of the quadratic equation inside the parentheses:
х⁴ - 3х³ + 8х² - 24х - 36 = 0
Using the factoring or synthetic division method, we can find the roots of this equation. Once we have the roots, we can determine the intervals where the inequality holds true.
First, let's break down the expression into its factors:
(х² - 2х)(х² - х - 6)(х² + 3х)
Now, let's multiply these factors together:
(х⁶ - 2х⁵ - 6х⁴ - х⁵ + 2х⁴ + 6х³ + 6х⁴ - 12х³ - 36х² + 3х⁵ - 6х⁴ - 18х³ + 3х⁴ - 6х³ - 18х²)
Simplify this expression:
х⁶ - 2х⁵ - 6х⁴ - х⁵ + 2х⁴ + 6х³ + 6х⁴ - 12х³ - 36х² + 3х⁵ - 6х⁴ - 18х³ + 3х⁴ - 6х³ - 18х²
Combine like terms:
х⁶ - 3х⁵ + 8х⁴ - 24х³ - 36х²
Now, we want to find the values of x where this expression is less than or equal to zero. To do this, we need to find the roots of this expression by setting it equal to zero:
х⁶ - 3х⁵ + 8х⁴ - 24х³ - 36х² = 0
Factor out an x²:
x²(х⁴ - 3х³ + 8х² - 24х - 36) = 0
Now, we need to find the roots of the quadratic equation inside the parentheses:
х⁴ - 3х³ + 8х² - 24х - 36 = 0
Using the factoring or synthetic division method, we can find the roots of this equation. Once we have the roots, we can determine the intervals where the inequality holds true.