Question

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

Answer 1

`x^2-10x`

Explanation:

A polynomial `P(x)` has a zero `x_0` if and only if `(x-x_0)` is a factor of `P(x)`. Using that, we can work backwards to make a polynomial with given zeros by multiplying each necessary factor of `(x-x_0)`.

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

`(x-0)(x-10) = x(x-10) = x^2-10x`

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.