Math Problem Statement
Solution
Let's break down the information step by step:
We are given a function:
Step 1: Identify Critical Points
First, we find the roots of , which are the values where the function equals zero:
Thus, the critical points where the sign of the function may change are at , , and .
Step 2: Sign Analysis
We now perform a sign analysis on the intervals determined by the critical points: , , , and .
-
Interval :
- , so is negative.
- , so is negative.
- , so is negative.
- The product is , so .
-
Interval :
- and , so is negative.
- , so is positive.
- , so is negative.
- The product is , so .
-
Interval :
- and , so is positive.
- , so is positive.
- , so is negative.
- The product is , so .
-
Interval :
- , so is positive.
- , so is positive.
- , so is positive.
- The product is , so .
Step 3: Analyze at Critical Points
- At , because of the factor .
- At , because of the factor .
- At , because of the factor .
Step 4: Determine Intervals for
From the sign analysis, on the following intervals:
Thus, the solution in interval notation is:
Would you like further details or have any questions?
Here are 5 related questions to expand on this topic:
- What is the significance of the critical points in this analysis?
- How can we graphically represent the sign changes of ?
- What is the difference between solving for versus ?
- How does the multiplicity of roots affect the behavior of polynomial functions?
- How can we apply this method to other polynomial functions?
Tip: When analyzing polynomials, the behavior at the roots (whether the function crosses the x-axis or just touches it) depends on the multiplicity of the roots.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Inequalities
Sign Analysis
Critical Points
Formulas
P(x) = x(x + 2)(x - 1)
Critical points from factors: x = -2, 0, 1
Theorems
Zero Product Property
Sign Chart Method
Suitable Grade Level
Grades 9-11