Find the complex zeros of the following polynomial function

Find the complex zeros of the following polynomial function Write f in f(x) = x^3 - 12x^2 + 49x - 58 The complex zeros of f are (Simplify your answer Type an exact answer, using radicals and i as need answers as needed) Use the complex zeros to factor f. f(x) = (Type your answer in factored form. Type an exact answer using radicals

Solution

f(x) = x^3 - 12 x^2 + 49 x - 58

possible rational zeros are factors of constant term divide by factors of lading term

possible rational zeros are +- { 1,2,29,58 } / {1}

real zero is x = 2

dividing the polynomial f(x) by x-2

x^3 - 12 x^2 + 49 x - 58 / x-2

we get x^2-10x+29

therefore setting x^2-10x+29 = 0 and solving for x by applying quadratic formula we get

x = {-b+- sqrt (b^2-4ac)}/2a

x = 5+2i

x = 5-2i

therefore, complex zeros of f are 5+2i , 5-2i

factored form is

(x-(5+2i)) (x-(5-2i) = (x - 5 - 2i )( x-5+2i)