Find the Roots (Zeros) f(x)=2x^3-4x^2-26x+52

Problem

ƒ(x)=2*x3−4*x2−26*x+52

Solution

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

2*x3−4*x2−26*x+52=0

2. Factor by grouping by splitting the polynomial into two pairs of terms.

(2*x3−4*x2)+(−26*x+52)=0

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

2*x2*(x−2)−26*(x−2)=0

4. Factor out the common binomial (x−2) from the expression.

(2*x2−26)*(x−2)=0

5. Factor out the constant 2 from the first binomial.

2*(x2−13)*(x−2)=0

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

x−2=0

x2−13=0

7. Solve for x in each equation to find all roots.

x=2

x2=13

x=±√(,13)

Final Answer

x=2,√(,13),−√(,13)

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