Identify the Zeros and Their Multiplicities f(x)=x^3-5x^2-4x+20

Problem

ƒ(x)=x3−5*x2−4*x+20

Solution

1. Group the terms of the polynomial to prepare for factoring by grouping.

ƒ(x)=(x3−5*x2)+(−4*x+20)

2. Factor out the greatest common factor from each group.

ƒ(x)=x2*(x−5)−4*(x−5)

3. Factor out the common binomial (x−5) from the expression.

ƒ(x)=(x2−4)*(x−5)

4. Apply the difference of squares formula to factor x2−4

ƒ(x)=(x−2)*(x+2)*(x−5)

5. Identify the zeros by setting each factor equal to zero and solving for x

x−2=0⇒x=2

x+2=0⇒x=−2

x−5=0⇒x=5

6. Determine the multiplicity of each zero by looking at the exponent of its corresponding factor. Since each factor appears exactly once, each zero has a multiplicity of 1.

Final Answer

x=2* (multiplicity 1),x=−2* (multiplicity 1),x=5* (multiplicity 1)

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