Simplifying Determinants to Finding Linear Factors - Homework Help

[Gregg]

Homework Statement

Express the determinant as a product of four linear factors.

<br /> \left(<br /> \begin{array}{ccc}<br /> 1 &amp; a &amp; a^3 \\<br /> 1 &amp; b &amp; b^3 \\<br /> 1 &amp; c &amp; c^3<br /> \end{array}<br /> \right)<br />

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.