Basic: $x^2 - 1$ and $-x^2 + 1$ have the same roots and therefore both could be written as $(x+1)(x-1).$ I guess that’s not right...

Question

I know they aren’t the same function but what’s the issue here? What am I not taking into account? Sorry for such a basic question, it just confuses me.

Answer 1

That product does not give the same function (try expanding it out, it gives one but not the other). You are trying to use the fact that "a function is a product of its roots".

But this is slightly inaccurate, rather the function may be written as $k(x+1)(x-1)$, where $k$ is any real number (in this case). You can see that all functions of this form have the same roots, and yet multiplying the roots does not necessarily give you back the same function.

In other words, if you have a polynomial with coefficients in the real numbers, then the product of roots is a factor of your function, but not necessarily the function itself.

Answer 2

$x^2-1$ and $7x^2-7$ both have the same roots.

One of them is $(x-1)(x+1)$ and the other is $7(x-1)(x+1).$

Two polynomials with the same roots, when factored, are constant multiples of each other.

In your example, the constant is $-1.$

(If you're talking about polynomials with real coefficients, then "the same roots" must be taken to mean the same complex roots, including, but not limited to, real roots. That is because $\mathbb C$ is the algebraic closure of $\mathbb R.$)