Question

How do you expand `(d-1/d)^5`?

Answer 1

Two ways to do it. Both using Newton's binomial theorem

Explanation:

Way 1 .

Apply directly Newton's theorem (alternate + and -)

`(d-1/d)^5=5C0d^5-5C1d^4·1/d^1+5C2d^3·1/d^2-5C3d^2·1/d^3+5C4d^1·1/d^4-5C5d^0·1/d^5` where

`mCn` are terms of Pascal triangle in the 5th row which are

5C0=1, 5C1=5, 5C2=10, 5C3=10, 5C4=5 and 5C5=1

And result in `d^5-5d^3+10d-10/d-5/d^3-1/d^5`

Way 2

Operate in `(d-1/d)^5=((d^2-1)/d)^5` and apply Newton's Theorem in numerator and denominator (which is simple `d^5`)