Question: Find the solution of the following polynomial inequality. Express your answer in interval notation…

Find the solution of the following polynomial inequality. Express your answer in interval notation.

\[ x(x + 3)^2(x - 4) < 0 \]

Answer 8 Points

Solution

Find the critical points by setting each factor equal to zero. \[ x = 0, \quad x = -3, \quad x = 4 \] These points divide the number line into intervals. \[ (-\infty, -3), \quad (-3, 0), \quad (0, 4), \quad (4, \infty) \] Determine the sign of the expression in each interval. \[ \begin{cases} x < -3: & (-)(+)^2(-) < 0 \\ -3 < x < 0: & (-)(+)^2(-) < 0 \\ 0 < x < 4: & (+)(+)^2(-) < 0 \\ x > 4: & (+)(+)^2(+) > 0 \\ \end{cases} \] The expression is negative in the intervals where the inequality holds. \[ (-\infty, -3) \cup (-3, 0) \cup (0, 4) \]