Question

How do you write a polynomial function given the real zeroes 3-i,5i and coefficient 1?

Answer 1

`y=x^4-6x^3+35x^2-150x+250`

Explanation:

Complex roots always come in pairs.

`3-i` will be a `0`, but so will `3+i`.

`5i` will be a `0`, but so will `-5i`.

This gives us the function

`y=(x-(3-i))(x-(3+i))(x-5i)(x-(-5i))`

`y=(x-3+i)(x-3-i)(x-5i)(x+5i)`

`y=((x-3)+i)((x-3)-i)(x^2-25i^2)`

`y=((x-3)^2-i^2)(x^2+25)`

`y=(x^2-6x+9+1)(x^2+25)`

`y=(x^2-6x+10)(x^2+25)`

`y=x^4-6x^3+35x^2-150x+250`

Notice how the function only has imaginary roots (never crosses the `x`-axis):

graph{x^4-6x^3+35x^2-150x+250 [-5, 8, -36, 300]}