Factorise..x^2-2+1/x^2-y^2
It helps to check your typing before you post.
Did you mean
x^2-2+1/x^2-y^2 , the way you typed it. Then there is nothing to do
or
(x^2-2+1)/(x^2-y^2) ?
or
(x^2-2x+1)/(x^2-y^2) ? , then
(x-1)^2 / ((x+y)(x-y))
heck, I wouldn't even rule out
(x^2-2xy+y^2)/(x^2-y^2) = (x-y)^2 / (x-y)(x+y) = (x-y)/(x+y)
To factorize the expression x^2 - 2 + 1/(x^2 - y^2), we can start by noticing that x^2 - y^2 is a difference of squares, which can be factored as (x + y)(x - y).
Next, we can rewrite the original expression as follows:
x^2 - 2 + 1/(x^2 - y^2)
Since (x^2 - y^2) can be factored as (x + y)(x - y), we can substitute it in:
x^2 - 2 + 1/[(x + y)(x - y)]
Now, let's find the common denominator by multiplying (x + y)(x - y) to the whole expression:
[(x^2 - 2)(x + y)(x - y) + 1]/[(x + y)(x - y)]
Next, let's expand the numerator:
(x^2 - 2)(x + y)(x - y) + 1
(x^3 - xy^2 + xy + y^3 - 2x^2 + 2y^2 + x - 2y + 1)
Now, we have the fully expanded expression:
(x^3 - xy^2 + xy + y^3 - 2x^2 + 2y^2 + x - 2y + 1)/[(x + y)(x - y)]
Although the expression is now fully expanded, it cannot be further factored as it does not have any common factors or quadratic terms.