Question

What is the complete factorization of the polynomial function over the set of complex numbers? f(x)=x^3+3x^2+16x+48

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Answer 1

Answer:
`(x+3)(x+4i)(x-4i)` 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+3x^2+16x+48=(x^3+3x^2)+(16x+48)`

Taking
`x^2` common from first group and the 16 from second group we get:

`f(x) = x^2(x+3)+16(x+3) = (x+3)(x^2+16)`

Now, to factor in complex from we have to break term
`x^2+16`

`f(x)= (x+3){x^2-(-4i)^2}`

As,
`i^2 = -1 , therefore (-4i)^2 = 16i^2 =-16`

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

On solving

`f(x) = (x+3)(x+4i)(x-4i)`

`(x+3)(x+4i)(x-4i)` is the required factorization of f(x).