Binomial expansion of modulus?

Question

How one can perform binomial expansion of modulus of a quantity. For example, can we expand |-1+x| ? If, yes how? Thanks

Answer 1

Two cases it depends on (-1+x) value

Case 1: (-1+x) is positive then write it as $(x-1)^n$ and use normal binomial theorem

Case 2: if (-1+x) is negative your answer is $-(x-1)^n$

Again if n is not a positive integer then you have to use

$$(1+x)^n=1+nx+ \frac{n(n-1)}{1 \cdot 2} x^2 + \frac{n(n-1)(n-2)}{1\cdot2\cdot3} x^3+\cdots$$

So then case 1 will be $$(-1)^n \bigg[\frac{n(n-1)}{1 \cdot 2}x^2 + \frac{n(n-1)(n-2)}{1\cdot2\cdot3} x^3+ \cdots \bigg]$$

Case 2 will be $$(-1)^{n+1}\bigg[1+nx+ \frac{n(n-1)}{1 \cdot 2}x^2 + \frac{n(n-1)(n-2)}{1\cdot2\cdot3} x^3+ \cdots\bigg]$$