Question

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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Answer 1

Answer:
`(x-5)(x+2i)(x-2i)` is the required factorization of f(x).

Explanation:

To factor the expression we must first group the terms and then take out common from these groups

`f(x)=x^3-5x^2+4x-20=(x^3-5x^2)+(4x-20)`

Taking
`x^2` common from first group and the 4 from second group we get:

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

Now, to factor in complex from we have to break term
`x^2+4`

`f(x)= (x-5){x^2-(-2i)^2}`

As,
`i^2 = -1 , therefore (-2i)^2 = 4i^2 =-4`

Also using identity
`a^2-b^2 =(a+b)(a-b)`

On solving

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

`(x+5)(x+2i)(x-2i)` is the required factorization of f(x).