Question

Expand ( x - 1/x^2)^4

Answer 1

Answer:

We want to expand the expression:

$(x - \frac{1}{x^2} )^4$

We can just do it by brute force, this is:

First, rewrite our expression as the product of two square factors:

$(x - \frac{1}{x^2} )^4 = (x - \frac{1}{x^2} )^2*(x - \frac{1}{x^2} )^2$

Now we can expand each one these two factors:

$(x - \frac{1}{x^2} )^2 = (x - \frac{1}{x^2} )*(x - \frac{1}{x^2} ) = x^2 + \frac{1}{x^4} -2*x*\frac{1}{x^2}$

That can be simplified to

$x^2 - \frac{2}{x} + \frac{1}{x^4}$

Now we can replace that in our original expression to get:

$(x^2 - \frac{2}{x} + \frac{1}{x^4})*(x^2 - \frac{2}{x} + \frac{1}{x^4})$

Now we can expand that last product, to get:

$(x^2)^2 + 2*(x^2)*(-\frac{2}{x} ) + 2*(x^2)*(\frac{1}{x^4}) + 2*(\frac{-2}{x})*(\frac{1}{x^4}) + (\frac{-2}{x} )^2 + (\frac{1}{x^4})^2$

We can simplify that to:

$x^4 - 4x + 2x^2 - \frac{4}{x^5} + \frac{4}{x^2} + \frac{1}{x^8}$

That is the expanded expression.