Simplest polynomial function is [tex]x^4-x^3-11x^2+9x+18.[/tex]
Solution:
Given data:
Zeroes are 3i, –1, 2.
3i is a complex root of the function.
If 3i is a zero of the polynomial then –3i is also a zero of the polynomial.
Therefore zeroes are 3i, –3i –1, 2.
By factor theorem,
If a is zero of the function, then (x – a) is a factor of the polynomial.
So, the factors are (x – 3i), (x + 3i), (x + 1), (x –2).
On multiplying the factors, we get the polynomial.
[tex](x - 3i) (x + 3i) (x + 1) (x -2)[/tex]
[tex]=(x^2+3ix-3ix+(3i)^2)(x^2-2x+x-2)[/tex]
[tex]=(x^2-9)(x^2-x-2)[/tex]
Since the value of [tex]i^2=-1[/tex]
[tex]=x^2(x^2-x-2)-9(x^2-x-2)[/tex]
[tex]=x^4-x^3-2x^2-9x^2+9x+18[/tex]
[tex]=x^4-x^3-11x^2+9x+18[/tex]
Simplest polynomial function is [tex]x^4-x^3-11x^2+9x+18.[/tex]
