# Compute the resultant of the polynomials $x^3-2x+4$ and $x^5+x^3-2x+4$

I know a little bit about the resultant of two polynomials but I couldn't find any example whatsoever. So I was wondering If you could illustrate the process of finding the resultant of two polynomials for $f = x^3-2x+4$ and $g = x^5+x^3-2x+4$. Also If you could give me another examples that you might think it could help me understant this concept would be great.
What I've tried so far was to see if f and g have common roots in which case the resultant would be 0, but that didn't work. After that I have tried to calculate the determinant of the Sylvester matrix but I'm not sure how to calculate that determinant: $$\begin{pmatrix} 1 & 0 & 1 & -1 & 0 & 4 &0 &0 \\ 0 & 1 & 0 & 1 & -1 & 0 & 4 & 0\\ 0 & 0 & 1 & 0 & 1 & -1 & 0 & 4\\ 1 & 0 & -2& 4& 0& 0 & 0 & 0\\ 0 & 1 & 0& -2 & 4& 0& 0 &0 \\ 0 & 0 & 1 & 0 & -2 & 4 & 0 &0 \\ 0& 0 & 0 & 1 & 0 & -2 & 4 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & -2 & 4 \end{pmatrix}$$

• – glS
Commented Apr 27, 2021 at 15:30

Your matrix would be correct if $g$ were $x^5+x^3-x^2+4$. But since $g$ is $x^5+x^3-2x+4$ instead, it should be $$\begin{pmatrix} 1 & 0 & 1 & 0 & -2 & 4 &0 &0 \\ 0 & 1 & 0 & 1 & 0 & -2 & 4 & 0\\ 0 & 0 & 1 & 0 & 1 & 0 & -2 & 4\\ 1 & 0 & -2& 4& 0& 0 & 0 & 0\\ 0 & 1 & 0& -2 & 4& 0& 0 &0 \\ 0 & 0 & 1 & 0 & -2 & 4 & 0 &0 \\ 0& 0 & 0 & 1 & 0 & -2 & 4 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & -2 & 4 \end{pmatrix}$$ instead.
What this Sylvester's determinant means is a necessary and sufficient condition for both polynomials $$f(x)=x^3-2x+4$$ and $$g(x)=x^5 +x^3-2x+4$$ have common roots which is not the case because the roots of $$f$$ are $$-2$$ and $$1\pm i$$ and none of these three roots is a root of $$g$$.