Factor the determinant of a 3×3 matrix with powers of x

[takercena]

Homework Statement

| x       x^2   1  |
| x^2      x    1  |
| x^3    x^3   1  |

The answer is = x^2(1-x)^2(1+2x)

Homework Equations

none

The Attempt at a Solution

By factoring the matrix step by step.

Thanks :)

 

[Defennder]

What's the problem here? Just use co-factor expansion along any row or column.

 

[jeffreydk]

Take advantage of the fact that you have a column of ones in the third column...

$$\begin{vmatrix}<br /> x & x^2 & 1 \\<br /> x^2 & x & 1 \\<br /> x^3 & x^3 & 1<br /> \end{vmatrix}<br /> = <br /> \begin{vmatrix}<br /> x^2 & x \\<br /> x^3 & x^3 <br /> \end{vmatrix} -<br /> \begin{vmatrix}<br /> x & x^2 \\<br /> x^3 & x^3 <br /> \end{vmatrix} +<br /> \begin{vmatrix}<br /> x & x^2 \\<br /> x^2 & x <br /> \end{vmatrix}$$

So what does this end up being?

 

[takercena]

Sorry for the question. I was confused by the question a moment ago because of language barrier.