I have a degree 6 polynomial with coefficients in terms of $a$ that I want to calculate the discriminant of. Is there an easy way to do this like an online calculator? The issue I am having is that searches I have done have only yielded online calculators that won't accept variable coefficients.
To be clear, the result I am looking for is the discriminant as a function of the coefficients of my degree 6 polynomial.
The answer for degree 6 polynomials takes a long time to find. Wolfram Alpha (https://www.wolframalpha.com/) produces several terms before the computation time is exceeded. You can type in "discriminant a x^3 + b x^2 + c x + d" as the query for a cubic, for example. Do you have a more specific question about the discriminant in this case, or is it just something out of interest?
A general formula for the discriminant of a univariate polynomial of degree $n$ is $\Delta(f) = (-1)^{\frac{n(n-1)}{2}} a_n^{2n - 2} \prod_{i < j} (x_i - x_j)^2$ where $a_n \not= 0 $ is the coefficient of $x^n$ in $f$ and $x_1, \ldots x_n$ are the roots of $f$.
This is the Product Formula, equation (1.23) in "Discriminants, Resultants and Multidimensional Determinants" by I.M. Gelfand et al., Birkhaueser, 1994, Chapter 12, page 403.
In Mathematica v 12, it is given by the Discriminant[poly, var] function. MAPLE has a similar function, discrim(p, x). For a quadratic, it finds $b^2 - 4ac$ for example.
For a univariate polynomial of degree 6, $a1 x^6 + a2 x^5 + \cdots + a7$, the discriminant begins: $a2^2 a3^2 a4^2 a5^2 a6^2 - 4 a1 a3^3 a4^2 a5^2 a6^2 - 4 a2^3 a4^3 a5^2 a6^2 + 18 a1 a2 a3 a4^3 a5^2 a6^2 - 27 a1^2 a4^4 a5^2 a6^2 - 4 a2^2 a3^3 a5^3 a6^2 + 16 a1 a3^4 a5^3 a6^2 + 18 a2^3 a3 a4 a5^3 a6^2 - 80 a1 a2 a3^2 a4 a5^3 a6^2 - \cdots$