Simplifying Determinants to Finding Linear Factors - Homework Help

[Gregg]

Homework Statement

Express the determinant as a product of four linear factors.

[itex]
\left(
\begin{array}{ccc}
1 & a & a^3 \\
1 & b & b^3 \\
1 & c & c^3
\end{array}
\right)
[/itex]

I'm sure that the only way to do this without hurting yourself is to operate on the determinant and take factors out. It just doesn't move. I can't see the pattern, I can't see how to get factors out of it. I add rows, columns everything.

 

[rock.freak667]

Expand along the first row and then just try to factorize it.

 

[Gregg]

Expand along the first row and then just try to factorize it.

Can't do it. It's hard to show my working on latex buts there's pages of attempts. x^3 I just can't do.

 

[rock.freak667]

Try row reducing once, with R₂-R₁ and R₃-R₁. Then expand along row one.


then use this identity and see if it helps a³-b³=(a-b)(a²+ab+b²)

 

[Gregg]

Try row reducing once, with R₂-R₁ and R₃-R₁. Then expand along row one.


then use this identity and see if it helps a³-b³=(a-b)(a²+ab+b²)

Excellent thanks, that's the identity I needed.