Convert to Interval Notation x^3-2x^2-3x<0

Problem

x3−2*x2−3*x<0

Solution

1. Factor out the greatest common factor, which is x

x*(x2−2*x−3)<0

2. Factor the quadratic expression inside the parentheses by finding two numbers that multiply to −3 and add to −2

x*(x−3)*(x+1)<0

3. Identify the critical points by setting each factor equal to zero.

x=0

x=3

x=−1

4. Test the intervals created by the critical points (−∞,−1) (−1,0) (0,3) and (3,∞) to see where the inequality holds true.
   For x=−2 (−2)*(−5)*(−1)=−10<0 (True)
   For x=−0.5 (−0.5)*(−3.5)*(0.5)=0.875<0 (False)
   For x=1 (1)*(−2)*(2)=−4<0 (True)
   For x=4 (4)*(1)*(5)=20<0 (False)

5. Combine the intervals where the expression is negative.

(−∞,−1)∪(0,3)

Final Answer

x3−2*x2−3*x<0⇒(−∞,−1)∪(0,3)

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