what is the determinant of

Question

I'm trying to solve my text book's determinant problem. the determinant is:

| 1  x  x^3 |
| 1  y  y^3 |
| 1  z  z^3 |

i have to prove that this determinant equals to (x+y+z)(x-y)(y-z)(z-y)

Answer 1

\begin{align*} \begin{vmatrix}1&x&x^3 \\ 1&y&y^3\\1&z&z^3\end{vmatrix}&=\begin{vmatrix}1&x&x^3 \\ 0&y-x&y^3-x^3\\0&z-x&z^3-x^3\end{vmatrix}\\ &=\begin{vmatrix}y-x&(y-x)(y^2+yx+x^2)\\z-x&(z-x)(z^2+zx+x^2)\end{vmatrix}\\ &=(y-x)(z-x)\begin{vmatrix}1&y^2+yx+x^2\\1&z^2+zx+x^2\end{vmatrix}\\ &=(y-x)(z-x)(z^2+zx-y^2-yx)\\ &=(y-x)(z-x)[(z-y)(z+y)+x(z-y)]\\ &=(y-x)(z-x)(z-y)(z+y+x)\\ &=(x+y+z)(x-y)(y-z)(z-x) \end{align*}

Answer 2

Expanding along the left column gives $$yz^3-y^3z-xz^3+x^3z+xy^3-x^3y$$. If you look at this, you’ll notice that it’s just $$(x-y)(y-z)(z-x)(x+y+z)$$ which i suspect is what you meant.