Question

How do you factor `x^(-1/2) - 2x^(1/2) + x^(3/2)`?

Answer 1

`x^(-1/2)(x-1)^2=(x-1)^2/x^(1/2)=(x^(1/2)(x-1)^2)/x`

Explanation:

Normally, when dealing with a trinomial, we're looking for something that looks like:

`ax^2+bx+c`

and our question doesn't have that! What it does have, however, if we compare the `c` term in what we are normally looking for to the `x^(-1/2)` in the question, the difference is the `(-1/2)` exponent. And, in fact, all the terms in the question have that same difference between what we expect over what we have. So let's first factor out a `x^(-1/2)` term:

`x^(-1/2)-2x^(1/2)+x^(3/2)`

`x^(0-1/2)-2x^(1- 1/2)+x^(4/2-1/2)`

`x^0 xx x^(-1/2)-2x xx x^(-1/2)+x^2 xx x^(-1/2)`

`x^(-1/2)(1-2x+x^2)`

and now we can factor in the normal way:

`x^(-1/2)(x^2-2x+1)`

`x^(-1/2)(x-1)^2=(x-1)^2/x^(1/2)=(x^(1/2)(x-1)^2)/x`