Question

How do you write a polynomial function given the real zeroes 1,-4, 5 and coefficient 1?

Answer 1

The simplest such polynomial is:

`f(x) = (x-1)(x+4)(x-5) = x^3-2x^2-19x+20`

Explanation:

Convert the zeros into linear factors to find:

`f(x) = (x-1)(x+4)(x-5) = x^3-2x^2-19x+20`

Any polynomial in `x` with these zeros will be a multiple (scalar or polynomial) of this `f(x)`.

Answer 2

Begin from the factored form to find the desired polynomial to be
`f(x) = x^3 -2x^2 - 19x + 20`

Explanation:

An easy way of generating a polynomial with a given set of zeros is to begin with the factored form. A polynomial `f(x)` has `a` as a zero if and only if `(x-a)` is a factor of `f(x)`.

Using this, we can construct the desired polynomial as follows:

`f(x) = (x-1)(x-(-4))(x-5)`

`=(x-1)(x+4)(x-5)`

`=x^3 -2x^2 - 19x + 20`

Note that we could multiply by a constant to give `x^3` a different coefficient, however this method naturally produces the coefficient of the highest power as `1`.