Answer:
[tex]f(x)=x^4-8x^3-13x^2-32x-68[/tex]
Step-by-step explanation:
We are given that the roots of the polynomial are 2i and 4-i.
Since, the given roots are complex roots and we know that, 'Complex roots always occurs in pairs'.
So, we get that,
The roots of the given polynomial are 2i, -2i, 4-i and 4+i.
i.e. [tex]f(x)=(x-2i)(x+2i)(x-(4-i))(x-(4+i))[/tex]
i.e. [tex]f(x)=(x^2-4i^2)(x^2-(4+i)x-(4-i)x-(4^2-i^2))[/tex]
i.e. [tex]f(x)=(x^2+4)(x^2-4x-ix-4x+ix-16+i^2))[/tex]
i.e. [tex]f(x)=(x^2+4)(x^2-8x-16-1))[/tex]
i.e. [tex]f(x)=(x^2+4)(x^2-8x-17))[/tex]
i.e. [tex]f(x)=x^4-8x^3-17x^2+4x^2-32x-68[/tex]
i.e. [tex]f(x)=x^4-8x^3-13x^2-32x-68[/tex]
Hence, the fourth degree polynomial having roots 2i, 4-i is [tex]f(x)=x^4-8x^3-13x^2-32x-68[/tex].
