How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where $f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$
and here $\Delta$ is the discriminant. The discriminant is what defines the nature of the roots (if they'll be real or complex, depending on whether $\Delta>0$ or $\Delta<0$).
How do you determine the discriminant of a cubic polynomial and higher polynomials?
In regards to quadratic polynomials: $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ is found by completing the square, which can be done to all quadratic polynomials, and then this formula is applied to find the roots.
But for cubic polynomials, quatric polynomials, degree $7$ polynomials, etc,…, how do you find the roots? (Also how do you determine the discriminant?)
If you have a cubic polynomial, sometimes you can factor it and make it a quadratic polynomial multiplied by another term: $$9t^3-18t^2+6= 3t(3t^2-6t+2)$$ but there are cases when you can't simplify it like this, right?
So I am just curious about how to find the roots/discriminant in higher degree polynomials.
Best Answer
If a polynomial is given numerically, (coefficients $a_0..a_n$ are given), the resultant method can be used to get at a numerical value of the discriminant. The coefficients of the polynomial and its derivative are put in a (n+2)- squared Sylvester matrix. Then the determinant is the discriminant wanted. Where writing out the discriminant of a matrix containing symbols is prohibitive, the discriminant can be calculated swiftly numerically using existing matrix packages in $O(N^3)$ time.
It is very nicely explained in
http://www2.math.uu.se/~svante/papers/sjN5.pdf
This is example 4.7
If $f(x) = ax^4 + bx^3 + cx^2 + dx + e$, then Theorem 3.3 yields
\begin{align} \begin{array}{| ccccccc |} & a & b & c & d & e & 0 & 0 &\\ & 0 & a & b & c & d & e & 0 \\ & 0 & 0 & a & b & c & d & e \\ & 4a& 3b& 2c& d & 0 & 0 & 0 \\ & 0 & 4a& 3b& 2c& d& 0 & 0 \\ & 0 & 0 & 4a& 3b& 2c& d & 0 \\ & 0 & 0 & 0 & 4a& 3b& 2c& d \\ \end{array} \end{align}
= $b^2c^2d^2 - 4b^2c^3 e - 4b^3d^3 + 18b^3cde - 27b^4e^2 - 4ac^3d^2 + 16ac^4e + 18abcd^3 - 80abc^2de - 6ab^2d^2e + 144ab^2ce^2 - 27a^2d^4 + 144a^2cd^2e - 128a^2c^2e^2 - 192a^2bde^2 + 256a^3e^3$