Question

Factor each of the following as completely as possible over the reals? a) `a^3 (c-b) + b^3 (a-c) + c^3 (b-a)` b) `a^4 (c-b) + b^4 (a-c) + c^4 (b-a)`

Factor each of the following as completely as possible over the reals?
a) `a^3 (c-b) + b^3 (a-c) + c^3 (b-a)`
b) `a^4 (c-b) + b^4 (a-c) + c^4 (b-a)`

Answer 1

Factor each of the following as completely as possible over the reals?
a) `a^3 (c-b) + b^3 (a-c) + c^3 (b-a)`
b) `a^4 (c-b) + b^4 (a-c) + c^4 (b-a)`

Ans

a) `a^3 (c-b) + b^3 (a-c) + c^3 (b-a)`

`=a^3 (c-b) + b^3 a-b^3c + c^3 b-c^3a`

`=a^3 (c-b) -c^3a+ b^3 a + c^3 b-b^3c`

`=a^3 (c-b) -a(c^3- b^3)+bc( c^2- b^2)`

`=a^3 (c-b) -a(c- b)(c^2+cb+b^2)+bc( c^2- b^2)`

`=(c-b)(a^3 -a(c^2+cb+b^2)+bc( c+b))`

`=(c-b)(a^3 -ac^2-acb-ab^2+bc^2+b^2c)`

`=(c-b)(a^3 -ac^2-acb+bc^2-ab^2+b^2c)`

`=(c-b)(a(a^2 -c^2)-cb(a-c)-b^2(a-c))`

`=(c-b)(a-c)(a(a+c)-cb-b^2)`

`=(c-b)(a-c)(a^2+ac-cb-b^2)`

`=(c-b)(a-c)(a^2-b^2+ac-cb)`

`=(c-b)(a-c)((a-b)(a+b)+c(a-b))`

`=(c-b)(a-c)(a-b)(a+b+c)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

b) `a^4 (c-b) + b^4 (a-c) + c^4 (b-a)`

`=a^4 (c-b) + b^4 a-b^4c + c^4 b-c^4a`

`=a^4 (c-b)-c^4a + b^4 a+ c^4 b-b^4c`

`=a^4 (c-b)-a(c^4 - b^4)+bc( c^3-b^3)`

`=a^4 (c-b)-a(c - b)(c+b)(c^2+b^2)+bc( c-b)(c^2+cb+b^2)`

`=( c-b)(a^4 -a(c+b)(c^2+b^2)+bc(c^2+cb+b^2))`

`=( c-b)(a^4 -ac^3-abc^2-ab^2c-ab^3+bc^3+c^2b^2+b^3c)`

`=( c-b)(a^4 -ac^3-abc^2+bc^3-ab^2c+c^2b^2-ab^3+b^3c)`

`=( c-b)(a(a^3 -c^3)-bc^2(a-c)-b^2c(a-c)-b^3(a-c))`

`=( c-b)(a-c)(a(a^2 +ac+c^2)-bc^2-b^2c-b^3)`

`=( c-b)(a-c)(a^3 +a^2c+ac^2-bc^2-b^2c-b^3)`

`=( c-b)(a-c)(a^3-b^3 +a^2c-b^2c+ac^2-bc^2)`

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

`=( c-b)(a-c)(a-b)(a^2+ab+b^2 +ca+cb+c^2)`