Algebraic Factorization/ Algebraic identity

oldman

Hi,

Are there any one of you who has a good collection of algebraic factorization/ algebraic identity in 3 or more variables. I have a lot of problem to manipulate inequality problems because I don't have a lot algebraic identity in my deposit. THanks in advance.........

**Posted:** Wed Jul 20, 2005 6:20 pm

_________________
Something certain is that everybody converges to death.

kunny

O.K. Here is a problem.

Factorize the following equations.

(1) $(a+b)(b+c)(c+a)+abc$

(2) $(a-b)^3+(b-c)^3+(c-a)^3$

(3) $a^4+b^4+c^4-2a^2b^2-2b^2c^2-2c^2a^2$

Source:1990 Meiji University of Pharmaceuitical

**Posted:** Wed Jul 20, 2005 7:26 pm

TomciO

We can use the identy $x^3 + y^3 + z^3 = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx) + 3xyz$ to solve the second one:
$(a-b)^3 + (b-c)^3 + (c-a)^3 = 3(a-b)(b-c)(c-a)$

**Posted:** Thu Jul 21, 2005 3:36 am

TomciO

And for the first one:
$(a+b)(b+c)(c+a) + abc = (a+b+c)(ab+bc+ca)$
It isn't hard to guess...

**Posted:** Thu Jul 21, 2005 3:47 am

shyong

for the third one we have

$a^4-2a^2(b^2+c^2)+b^4-2b^2c^2+c^4 =a^4-2a^2(b^2+c^2)+(b^2-c^2)^2 =a^4-2a^2(b^2+c^2)+(b+c)^2(b-c)^2 =(a^2-(b+c)^2)(a^2-(b-c...$

**Posted:** Sat Jul 23, 2005 10:02 am

_________________
Life is complex: it has both real and imaginary components Mr. Green

kunny

**Posted:** Sat Jul 23, 2005 3:32 pm

kunny

**Posted:** Sat Jul 23, 2005 3:33 pm