Identify the Zeros and Their Multiplicities 2x^2(x-1)(x+2)^3(x^2+1)^2

Problem

2*x2*(x−1)*(x+2)3*(x2+1)2

Solution

1. Set the expression to zero to find the roots of the polynomial.

2*x2*(x−1)*(x+2)3*(x2+1)2=0

2. Apply the Zero Product Property by setting each distinct factor equal to zero.

x2=0

x−1=0

x+2=0

x2+1=0

3. Solve for x for each factor to identify the real and complex zeros.

x=0

x=1

x=−2

x2=−1⇒x=±i

4. Identify the multiplicity of each zero by looking at the exponent of its corresponding factor.
   The factor x2 has exponent 2 so x=0 has multiplicity 2
   The factor (x−1) has exponent 1 so x=1 has multiplicity 1
   The factor (x+2)3 has exponent 3 so x=−2 has multiplicity 3
   The factor (x2+1)2 can be written as ((x−i)*(x+i))2 so x=i and x=−i both have multiplicity 2

Final Answer

x=0* (multiplicity 2), *x=1* (multiplicity 1), *x=−2* (multiplicity 3), *x=±i* (multiplicity 2)

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