Find the Roots (Zeros) f(x)=5x^3-x^2-5x+1

Problem

ƒ(x)=5*x3−x2−5*x+1

Solution

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

5*x3−x2−5*x+1=0

2. Group the terms to prepare for factoring by grouping.

(5*x3−x2)+(−5*x+1)=0

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

x2*(5*x−1)−1*(5*x−1)=0

4. Factor out the common binomial (5*x−1) from the expression.

(x2−1)*(5*x−1)=0

5. Apply the difference of squares formula to factor x2−1

(x−1)*(x+1)*(5*x−1)=0

6. Set each factor to zero using the zero product property to solve for x

x−1=0⇒x=1

x+1=0⇒x=−1

5*x−1=0⇒x=1/5

Final Answer

x=1,−1,1/5

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