What is the complete factorization of the polynomial function over the set of complex numbers?

f (x) = x3-5x2+4x-20

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

Step-by-step explanation:

Given that

f (x) = x3-5x2+4x-20

Grouping first two terms together and second set together we have

f (x) = x^2 (x-5) + 4 (x-5)

= (x^2+4) (x-5)

These are the factors without complex numbers of f (x)

Equate each factor to 0 to get

x-5 = 0 or x^2+4=0

x = 5 or x = square root of - 4

i. e. x=5 or x = 2i or x = -2i (since square root of negative number is imaginary)

Factorisation is

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