To solve the inequality [tex]\( 3x^3 + 9x^2 - 12x < 0 \)[/tex] using the end behavior of the graph, follow these steps:
1. Factor the polynomial: Factor the expression [tex]\( 3x^3 + 9x^2 - 12x \)[/tex].
[tex]\[
3x^3 + 9x^2 - 12x = 3x(x^2 + 3x - 4)
\][/tex]
Next, factor the quadratic [tex]\( x^2 + 3x - 4 \)[/tex].
[tex]\[
x^2 + 3x - 4 = (x + 4)(x - 1)
\][/tex]
So, the factored form of the original inequality is:
[tex]\[
3x(x + 4)(x - 1) < 0
\][/tex]
2. Find the critical points: Identify the roots of the polynomial, which are the points where the expression equals zero.
[tex]\[
3x(x + 4)(x - 1) = 0
\][/tex]
The roots are [tex]\( x = 0 \)[/tex], [tex]\( x = -4 \)[/tex], and [tex]\( x = 1 \)[/tex].
3. Determine the intervals: Use the roots to divide the real number line into intervals:
[tex]\[
(-\infty, -4), (-4, 0), (0, 1), (1, \infty)
\][/tex]
4. Test the sign of the polynomial in each interval: Choose a test point from each interval and determine the sign of the expression [tex]\( 3x(x + 4)(x - 1) \)[/tex].
- For [tex]\( x \in (-\infty, -4) \)[/tex]:
Choose [tex]\( x = -5 \)[/tex]:
[tex]\[
3(-5)((-5) + 4)((-5) - 1) = 3(-5)(-1)(-6) = -90 \quad (\text{negative})
\][/tex]
- For [tex]\( x \in (-4, 0) \)[/tex]:
Choose [tex]\( x = -2 \)[/tex]:
[tex]\[
3(-2)((-2) + 4)((-2) - 1) = 3(-2)(2)(-3) = 12 \quad (\text{positive})
\][/tex]
- For [tex]\( x \in (0, 1) \)[/tex]:
Choose [tex]\( x = 0.5 \)[/tex]:
[tex]\[
3(0.5)((0.5) + 4)((0.5) - 1) = 3(0.5)(4.5)(-0.5) = -3.375 \quad (\text{negative})
\][/tex]
- For [tex]\( x \in (1, \infty) \)[/tex]:
Choose [tex]\( x = 2 \)[/tex]:
[tex]\[
3(2)(2 + 4)(2 - 1) = 3(2)(6)(1) = 36 \quad (\text{positive})
\][/tex]
5. Combine the results: The inequality [tex]\( 3x(x + 4)(x - 1) < 0 \)[/tex] is satisfied where the polynomial is negative.
From our test points, the polynomial is negative in the intervals:
[tex]\[
(-\infty, -4)\quad \text{and}\quad (0, 1).
\][/tex]
Therefore, the solution to the inequality [tex]\( 3x^3 + 9x^2 - 12x < 0 \)[/tex] is:
[tex]\[
\boxed{(-\infty, -4) \cup (0, 1)}
\][/tex]
This matches the result [tex]\( \text{Union(Interval.open(-oo, -4), Interval.open(0, 1))} \)[/tex].
