# How do you find a polynomial function that has zeros 0, 10?

Jan 10, 2017

#### Answer:

${x}^{2} - 10 x$

#### Explanation:

A polynomial $P \left(x\right)$ has a zero ${x}_{0}$ if and only if $\left(x - {x}_{0}\right)$ is a factor of $P \left(x\right)$. Using that, we can work backwards to make a polynomial with given zeros by multiplying each necessary factor of $\left(x - {x}_{0}\right)$.

As our desired polynomial has $0$ and $10$ as zeros, it must have $\left(x - 0\right)$ and $\left(x - 10\right)$ as factors. Multiplying these, we get

$\left(x - 0\right) \left(x - 10\right) = x \left(x - 10\right) = {x}^{2} - 10 x$

This is a polynomial of least degree which has $0$ and $10$ as zeros. Note that multiplying this by any other polynomial or constant will also result in a polynomial with $0$ and $10$ as zeros.