Question

If a polynomial function with rational coefficients has the zeros -5 and i, what are the additional zeros?

Answer 1

Additional zero is `-i` and polynomial function is `x^3+5x^2+x+5`

Explanation:

A polynomial function whose zeros are `alpha`, `beta`, `gamma` and `delta` and multiplicities are `p`, `q`, `r` and `s` respectively is

`(x-alpha)^p(x-beta)^q(x-gamma)^r(x-delta)^s`

It is apparent that the highest degree of such a polynomial would be `p+q+r+s`.

In the given case as two zeros are `-5` and `i` and coefficients are rational, assuming multiplicity of each to be just `1`,

we will have to have additionally a zero `-i` (being complex conjugate of `i`) and hence polynomial is

`(x-(-5))(x-i)(x-(-i))`

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

= `(x+5)(x^2+1)`

= `x^3+5x^2+x+5`