To solve the inequality [tex]\( x^3 + x^2 - 26x + 24 \geq 0 \)[/tex], we need to find the values of [tex]\( x \)[/tex] for which this expression is greater than or equal to zero. Here is a step-by-step solution:
1. Identify the Inequality: We need to solve [tex]\( x^3 + x^2 - 26x + 24 \geq 0 \)[/tex].
2. Factor the Polynomial (if possible): Factoring is a common technique used to solve polynomial inequalities. However, in this case, we will utilize the result directly:
3. Determine the Roots:
The polynomial has roots which divide the real line into intervals. These roots are obtained as critical points where the inequality may change its sign.
4. Evaluate Intervals Between Roots:
Once we have the roots, we need to test the intervals between these roots to determine where the polynomial is non-negative.
Upon executing this process, it is found that:
For [tex]\( x \)[/tex] in the interval [tex]\([-6, 1]\)[/tex] and [tex]\( x \)[/tex] in the interval [tex]\([4, \infty)\)[/tex], the function [tex]\( x^3 + x^2 - 26x + 24 \)[/tex] is greater than or equal to zero. These intervals are determined by evaluating the polynomial on sample points within the intervals defined by the roots and checking the sign of the results.
Thus, our solution to the inequality [tex]\( x^3 + x^2 - 26x + 24 \geq 0 \)[/tex] in interval notation, combining these intervals, is:
[tex]\[
\boxed{[-6, 1] \cup [4, \infty)}
\][/tex]
This means that for any [tex]\( x \)[/tex] value within the intervals [tex]\([-6, 1]\)[/tex] or [tex]\([4, \infty)\)[/tex], the polynomial expression will be greater than or equal to zero.
