Factorise..(x+y-z)^3 +(x-y+z)^3 -8x^3
-6x(x+y-z)(x-y+z)
hint: (x+(y-z))(z-(y-z)) = x^2 - (y-z)^2
(x+(y-z))^3 + (x-(y-z))^3 = (x+y-z)(x^2 - (y-z)(y+z) + (y-z)^2)
To factorize the given expression, let's break it down step by step.
First, let's look at the expression that we need to factorize:
(x+y-z)^3 + (x-y+z)^3 - 8x^3
To factorize this expression, we can use the formula for factoring the sum of cubes:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
Now, let's apply this formula to factorize the first two terms, (x+y-z)^3 + (x-y+z)^3, individually:
(x+y-z)^3 can be written as (x + y - z)(x^2 + y^2 + z^2 + 2xy - 2xz - 2yz)
(x-y+z)^3 can be written as (x - y + z)(x^2 + y^2 + z^2 - 2xy + 2xz - 2yz)
So, the first two terms of the expression can be written as:
(x+y-z)^3 + (x-y+z)^3 = (x + y - z)(x^2 + y^2 + z^2 + 2xy - 2xz - 2yz) + (x - y + z)(x^2 + y^2 + z^2 - 2xy + 2xz - 2yz)
Now, let's simplify the third term, -8x^3:
-8x^3 can be factored out as -8(x^3)
Now, we can rewrite the full expression as:
(x + y - z)(x^2 + y^2 + z^2 + 2xy - 2xz - 2yz) + (x - y + z)(x^2 + y^2 + z^2 - 2xy + 2xz - 2yz) - 8(x^3)
And this is the factored form of the given expression.