What would a fast way to find the root (where equation becomes 0) of the polynomial equation? We all know the easy way to find the solution for quadratic equation, but not for others..
Would this involve linear algebra? Or other methods?
There is a very well known formula for the quadratics (I see that you know it). I assume someone may have asked a similar question on this website so I wouldn't be surprised to see a "duplicate" coming up in the comments...
But for degree $3$ and $4$ there are formulas, known as Cardano's formulas (other mathematicians have found these formulas too, but Cardano's name is usually stuck on them) which give the solutions of a polynomial equation as a function of its coefficients. The modern proofs involve Galois Theory, which is in some sense a mix of group theory and field theory, so I would put it as "a little linear algebra but mostly other methods".
For degree $n \ge 5$, it is known that there are no formulas involving either addition, subtraction, multiplication, division, radicals, or any combination/composition of them as a function of the coefficients. This is a very deep result, because it means that for polynomials of degree $5$ or more we cannot expect to find the roots in an easy manner at all.
As little satisfaction we can have for these polynomials, there exists numerical methods if we wish for instance to compute the roots over $\mathbb R$ or $\mathbb C$. Over other fields we are pretty much screwed in general, unless the polynomials we look at are particularly pretty and we have a clever way of "guessing" the roots using theory.
Hope that helps,