Factorisation of a determinant

[escobarro]

Hey
for this question:

\(\displaystyle 24.\, \mbox{ Establish the following factorizations:}\)

. . . . .\(\displaystyle \left|\, \begin{array}{ccc}1&1&1\\a&b&c\\a^3&b^3&c^3 \end{array}\, \right|\, =\, (b\, -\, c)\, (c\, -\, a)\, (a\, -\, b)\, (a\, +\, b\, +\, c)\)

It is asking for proof of finding the determinant and factorising to the given expression I think?
So, I found the determinant (through row expansion) but do not know how to factorise: bc^3+ca^3+ab^3-ba^3-cb^3-ac^3
this determinant should be correct as I factorised with a calculator to get the correct answer

Thanks

 

[stapel]

\(\displaystyle 24.\, \mbox{ Establish the following factorizations:}\)

. . . . .\(\displaystyle \left|\, \begin{array}{ccc}1&1&1\\a&b&c\\a^3&b^3&c^3 \end{array}\, \right|\, =\, (b\, -\, c)\, (c\, -\, a)\, (a\, -\, b)\, (a\, +\, b\, +\, c)\)

It is asking for proof of finding the determinant and factorising to the given expression I think?

It's asking you to show that the two sides are equal. When you multiplied out the right-hand side, did you get an expression equal to what you got for the determinant? Then you've "established" that the factorization on the right-hand side is equal to the determinant on the left-hand side.

 

[escobarro]

It's asking you to show that the two sides are equal. When you multiplied out the right-hand side, did you get an expression equal to what you got for the determinant? Then you've "established" that the factorization on the right-hand side is equal to the determinant on the left-hand side.

thanks!
yes, I did get the same expression after expanding the right hand side but for the next question:

the answer is:

so it looks like we are supposed to know how to factorise these?

 

[escobarro]

I found a way of doing it using long division of polynomials!