Question

How do you write a polynomial with zeros: -3,1,5,6?

Answer 1

For a zero, when `x` has that value the overall bracket (parenthesis) needs to sum to zero. For example `(x+3)=(-3+3)=0`. That means the solution is `(x+3)(x-1)(x-5)(x-6)`.

Explanation:

A 'zero' or 'root' of an equation is a point at which it crosses the `x`-axis: that is, the line `y=0` (or `f(x)=0`).

For that to happen, one of the factors must go to zero. If we take the example of `x=-3`, we need a factor that, when `x` has that value, will equal zero. If it's in the form `(x+a)` and we put in the value `x=-3`, the value of `a` required is `3`.

For a positive value of `x` such as `x=5`, the factor needs to be `(x-5)` so that it becomes zero when `x=5`.

Using this approach, over all we get:

`(x+3)(x-1)(x-5)(x-6)`

We could multiply this through to get an expression in `x^4` and all lower values, but it's probably best left in this factorized form.