Question

Solving a polynomial inequality: Solve the following inequality. (x+1)(x-3)²(x-8)<0 Write your answer as an interval or union of intervals. If there is no real solution, click on

Answer 1

To solve the polynomial inequality, we need to analyze the sign of each factor and find the intervals where the expression is negative. The solution to the inequality (x+1)(x-3)²(x-8)<0 is x < -1 or 3 < x < 8.

Step-by-step explanation:

To solve the inequality (x+1)(x-3)²(x-8)<0, we need to find the intervals where the expression is negative. We can do this by analyzing the sign of each factor and using the concept of interval notation.

The factors are (x+1), (x-3)², and (x-8). To determine the sign of each factor, we can use test points within the intervals. Let's consider the intervals x<-1, -1<x<3, and x>3:

1. For x<-1, a test point like x=-2 gives (x+1)<0, (x-3)²>0, and (x-8)<0. Therefore, the expression is negative in this interval.
2. For -1<x<3, a test point like x=0 gives (x+1)>0, (x-3)²<0, and (x-8)<0. Therefore, the expression is positive in this interval.
3. For x>3, a test point like x=4 gives (x+1)>0, (x-3)²>0, and (x-8)>0. Therefore, the expression is negative in this interval.

Based on the signs within each interval, we can conclude that the solution to the inequality is:

x < -1 or 3 < x < 8