Question

How do you write a polynomial in standard form given the zeros x=-1/2,0,4?

Answer 1

You multiply `x` minus the solution and you obtain `x^3-7/2x^2-2x=0`.

Explanation:

The solutions of a polynomial are those numbers that assigned to the `x` gives zero. So the easiest way to construct the polynomial from the solutions is to multiply together `x` minus the solution because you are sure that it becomes zero.
In our case we have

`x-(-1/2)`
`x-0`
`x-4`

It is clear that the three of them become zero when x is, respectively `-1/2`, `0` and `4`. If we multiply the three we have

`(x+1/2)x(x-4)`

we have a polynomial that will become zero for `-1/2`, `0` and `4` because these three numbers set as zero one of the three factors and the product for zero is zero, no matter what are the other factors.

Now it is just a multiplication to do to obtain the final result

`(x+1/2)x(x-4)`
`=(x+1/2)(x^2-4x)`
`=x^3-4x^2+1/2x^2-2x`
`=x^3-7/2x^2-2x`

and the corresponding equation is

`x^3-7/2x^2-2x=0`.