factorise
4(x-y)^3-(x-y)
first get rid of the (x-y):
(x-y)(4(x-y)^2-1)
Now you have a difference of two squares, so
(x-y) (2(x-y)-1) (2(x-y)+1)
...
To factorize the expression 4(x-y)^3-(x-y), we can observe that (x-y) appears in both terms. We can factor it out as a common factor:
4(x-y)^3 - (x-y) = (x-y)(4(x-y)^2 - 1)
Now, let's focus on the expression inside the parentheses, 4(x-y)^2 - 1. This is a difference of squares since we have (x-y)^2 with a coefficient of 4 and 1 being squared. We can factor it using the formula a^2 - b^2 = (a+b)(a-b):
4(x-y)^2 - 1 = (2(x-y))^2 - 1^2 = (2(x-y) - 1)(2(x-y) + 1)
Substituting this back into our original expression, we have:
4(x-y)^3 - (x-y) = (x-y)(4(x-y)^2 - 1) = (x-y)(2(x-y) - 1)(2(x-y) + 1)
So, the factored form of 4(x-y)^3 - (x-y) is (x-y)(2(x-y) - 1)(2(x-y) + 1).