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

Nov 15, 2016

Additional zero is $- i$ and polynomial function is ${x}^{3} + 5 {x}^{2} + x + 5$

#### Explanation:

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

${\left(x - \alpha\right)}^{p} {\left(x - \beta\right)}^{q} {\left(x - \gamma\right)}^{r} {\left(x - \delta\right)}^{s}$

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

$\left(x - \left(- 5\right)\right) \left(x - i\right) \left(x - \left(- i\right)\right)$

= $\left(x + 5\right) \left(x - i\right) \left(x + i\right)$

= $\left(x + 5\right) \left({x}^{2} + 1\right)$

= ${x}^{3} + 5 {x}^{2} + x + 5$