Factor \(ab^3 - a^3 b + bc^3 - b^3 c + ca^3 - c^3 a\).

I've gotten the factored version of the polynomial is -(a-b)(b-c)(c-a)(a+b+c), however I'm not really sure how to explain it. I know that we must begin with (a-b)(b-c)(c-a), but don't know where to go from there.

Thank you in advance.

grs75 Mar 14, 2021

#4**+3 **

*Gasp* this one is not easy.

Beginning with (a-b)(b-c)(c-a) is correct.

That will get us (ab-ac-b^2+bc)(c-a) which is ab^2-a^2b-ac^2-b^2c+bc^2.

And then we can see that, ab^2-a^2b-ac^2-b^2c+bc^2 times -(a+b+c) will get us ab^3-a^3b+bc^3-b^3c+ca^3-c^3a.

So, we combine this to the factor we already got: (a-b)(b-c)(c-a) and get -(a+b+c)(a-b)(b-c)(c-a), which is the same as -(a-b)(b-c)(c-a)(a+b+c)

CalTheGreat Mar 14, 2021

#5**+1 **

Hi Cal,

You have done some excellent expanding. Good work.

But

I was not asking anyone to show that the answer was correct.

I was asking for someone to explain how the answer could be arrived at, if I did not already know it.

I'm pretty sure that is what grs75 wants too.

Melody
Mar 15, 2021

#6**+1 **

Does this just depend on reasoning perhaps? I don't think there's anything complex, I simply got the answer by playing around but obviously that's not the correct way to solve it.

grs75
Mar 15, 2021

#8**+1 **

Oop yeah, I thought that grs75 already know how to get the (a-b)(b-c)(c-a) part.

In that case I'll gib some factoring tricks that can make this easier:

(a^b)^c = a^c*a^b

(a/b)^c=a^c/b^c

(a+b)^2=a^2+2ab+b^2

(a-b)^2=a^2-2ab+b^2

a^2-b^2=(a+b)(a-b)

(a+b)^3=a^3+3a^2b+3ab^2+b^3

a^3-b^3=(a-b)(a^2+a+b^2)

a^3+b^3=(a+b)(a^2-ab+b^2)

I'm sure that if we rearrange this SOMEHOW we'll get it, but apparently I'm dumb today

CalTheGreat Mar 15, 2021