# How do you form a polynomial function whose zeros, multiplicities and degrees are given: Zeros: -2, 2, 3; degree 3?

Nov 15, 2016

#### Answer:

Polynomial function is ${x}^{3} - 3 {x}^{2} - 4 x + 12$

#### 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 $p + q + r + s$.

As zeros are $- 2$, $2$ and $3$ and degree is $3$, it is obvious that multiplicity of each zero is just $1$.

Hence polynomial is $\left(x - \left(- 2\right)\right) \left(x - 2\right) \left(x - 3\right)$

= $\left(x + 2\right) \left(x - 2\right) \left(x - 3\right)$

= $\left({x}^{2} - 4\right) \left(x - 3\right)$

= ${x}^{3} - 3 {x}^{2} - 4 x + 12$