# What is the function with roots 2, 4 + i and 4 - i

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I am not sure if you want a function with the roots being ** only** 2, 4 + i and 4 - i or a function which has the roots 2, 4 + i and 4 - i.

As there can be an infinite number of functions that meet the latter condition, I am providing a function that has only 2, 4 + i and 4 - i as the roots.

f(x) = (x - 2)(x - (4 + i))(x - (4 - i))

=> f(x) = (x - 2)(x - 4 - i)(x - 4 + i)

=> f(x) = (x - 2)((x - 4)^2 - i^2)

use i^2 = -1

=> f(x) = (x - 2)((x - 4)^2 + 1)

open the brackets

=> f(x) = (x - 2)(x^2 + 16 - 8x + 1)

=> f(x) = (x - 2)(x^2 - 8x + 17)

=> f(x) = x^3 - 8x^2 + 17x - 2x^2 + 16x - 34

=> f(x) = x^3 - 10x^2 + 33x - 34

**The function with the roots 2, 4 + i and 4 - i is f(x) = x^3 - 10x^2 + 33x - 34**

Since the roots are 2, 4+i) and (4-i)

Then the factors are:

f(x) = (x-2)(x-(4+i) ( 4-(4-i)

Now we will open the brackets and determine the function f(x).

==> First we will open the bracktes ( x- (4+i) (x-(4-i)

==> (x^2 -8x +17 )

==> f(x) = (x-2)(x^2-8x+17)

==> f(x)= ( x^3 -8x^2 + 17x -2x^2 + 16x - 34

**==> f(x) = x^3 -10x^2 + 33x -34**