# Why do we need to include the factor $a_{n}^{2n-2}$ in the discriminant of a polynomial?

Question: Why do we need to include the factor $$a_{n}^{2n-2}$$ in the discriminant of a polynomial?

Here is the definition of the discriminant ($$\Delta$$) in terms of the roots $$r_1,r_2,...$$:

$$\Delta=a_{n}^{2n-2}\prod_{i for the polynomial $$a_nx^n+a_{n-1}x^{n-1}+...+a_0$$.

If the polynomial equation has non-real coefficients (in particular if $$a_n$$ is not real), then it is pointless to tell whether $$\Delta>0$$ and so in general I can't tell the number of real roots using this way for a polynomial equation with non-real coefficients.

However, for a polynomial equation with only real coefficients, even if we just consider $$\prod_{i, where $$a_{n}^{2n-2}$$ is not included, we still have:

$$\Delta >0, \text{if the number of complex roots} \equiv 0\mod 4$$ $$\Delta =0, \text{if there is a multiple root}$$ $$\Delta<0, \text{otherwise}$$

So, why isn't this the definition of discriminant? What difference does the factor $$a_{n}^{2n-2}$$ make in the non-real coefficients case?

Any help will be appreciated!

You can easily see that for a polynomial $$f(x)=a_nx^n+…+a_0$$ of degree $$n$$ $$Res(f,f’)=(-1)^{\frac {n(n-1)}2}a_n\Delta$$ where $$Res(f,f’)$$ stays for the resultant of the polynomials $$f$$ and its derivative $$f’$$. Since $$f$$ and $$f’$$ have the same leading coefficient, i.e. $$a_n$$, then $$Res(f,f’)\in a_n\Bbb Z[a_0,…,a_n]$$ and thus $$\Delta\in \Bbb Z[a_0,…,a_n]$$ that’s the reason to have the exponent in the definition of the discriminant: in this way the discriminant of a polynomial $$f$$ is always a integer polynomial in the coefficients of $$f$$!
• So the factor $a_{n}^{2n-2}$ is included to ensure the coefficients of the discriminant (as a polynomial) must be integers?