To solve the inequality [tex]\(x^3 + 2x^2 - 5x - 6 > 0\)[/tex], we need to determine the intervals where the polynomial is positive. Here is a detailed, step-by-step solution:
### Step 1: Find the Roots of the Polynomial
First, we need to find the roots (solutions) of the equation [tex]\(x^3 + 2x^2 - 5x - 6 = 0\)[/tex]. These roots are the points where the polynomial crosses the x-axis.
### Step 2: Factor the Polynomial (or find the roots)
Upon solving the polynomial equation [tex]\(x^3 + 2x^2 - 5x - 6 = 0\)[/tex], we find the roots are [tex]\(x = -3\)[/tex], [tex]\(x = -1\)[/tex], and [tex]\(x = 2\)[/tex].
### Step 3: Determine the Sign of the Polynomial in each Interval
The roots divide the number line into four intervals:
1. [tex]\(x < -3\)[/tex]
2. [tex]\(-3 < x < -1\)[/tex]
3. [tex]\(-1 < x < 2\)[/tex]
4. [tex]\(x > 2\)[/tex]
We need to test the sign of the polynomial [tex]\(x^3 + 2x^2 - 5x - 6\)[/tex] in each of these intervals.
- For [tex]\(x < -3\)[/tex]: Select a test point, say [tex]\(x = -4\)[/tex]. Substituting [tex]\(x = -4\)[/tex] into the polynomial:
[tex]\[
(-4)^3 + 2(-4)^2 - 5(-4) - 6 = -64 + 32 + 20 - 6 = -18
\][/tex]
The polynomial is negative in this interval.
- For [tex]\(-3 < x < -1\)[/tex]: Select a test point, say [tex]\(x = -2\)[/tex]. Substituting [tex]\(x = -2\)[/tex] into the polynomial:
[tex]\[
(-2)^3 + 2(-2)^2 - 5(-2) - 6 = -8 + 8 + 10 - 6 = 4
\][/tex]
The polynomial is positive in this interval.
- For [tex]\(-1 < x < 2\)[/tex]: Select a test point, say [tex]\(x = 0\)[/tex]. Substituting [tex]\(x = 0\)[/tex] into the polynomial:
[tex]\[
0^3 + 2(0)^2 - 5(0) - 6 = -6
\][/tex]
The polynomial is negative in this interval.
- For [tex]\(x > 2\)[/tex]: Select a test point, say [tex]\(x = 3\)[/tex]. Substituting [tex]\(x = 3\)[/tex] into the polynomial:
[tex]\[
3^3 + 2(3)^2 - 5(3) - 6 = 27 + 18 - 15 - 6 = 24
\][/tex]
The polynomial is positive in this interval.
### Step 4: Combine the Intervals
From the above steps, we can see that the polynomial [tex]\(x^3 + 2x^2 - 5x - 6\)[/tex] is greater than zero in the intervals:
- [tex]\(-3 < x < -1\)[/tex]
- [tex]\(2 < x < \infty\)[/tex]
Thus, the solution to the inequality [tex]\(x^3 + 2x^2 - 5x - 6 > 0\)[/tex] is:
[tex]\[
(-3 < x < -1) \cup (2 < x < \infty)
\][/tex]
