Question

Find a polynomial function that has the given zeros. (There are many correct answers.)? 7, 4 + root6, 4 − root6

Answer 1

One answer `(x-7)(x^2-8x+10)`.
There are, of course, infinitely many polynomials, but each one will have this as a factor.

Explanation:

If polynomial function vanishes at `x=a`, then `(x-a)` must be one of its factors.
Thus `(x-7)`, `(x-4-sqrt{6})` and `(x-4+sqrt{6})` are factors of the required polynomial.

The simplest polynomial that satisfies this is

`(x-7)(x-4-sqrt{6})(x-4+sqrt{6}) = (x-7){(x-4)-sqrt{6}}{(x-4)+sqrt{6}} = (x-7){(x-4)^2-6} = (x-7)(x^2-8x+16-6) = (x-7)(x^2-8x+10)`