# How do you write a polynomial in standard form given the zeros x=1, 2, and 3?

May 12, 2016

The simplest such polynomial would be $\textcolor{b l u e}{{x}^{3} - 6 {x}^{2} + 11 x - 6}$

#### Explanation:

If $1 , 2 , \mathmr{and} 3$ are zeros of the polynomial then the polynomial must contain (at least) factors:
$\textcolor{w h i t e}{\text{XXX}} \left(x - 1\right)$
$\textcolor{w h i t e}{\text{XXX}} \left(x - 2\right)$ and
$\textcolor{w h i t e}{\text{XXX}} \left(x - 3\right)$

If these are the only factors then our polynomial is
$\textcolor{w h i t e}{\text{XXX}} \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)$

To express this in standard form we need to expand by multiplying and make sure that the terms are listed in descending degree.