It's been like 20 years since I've done math like this. Can someone smarter than me remind me why this is wrong?
Cancelation between a numerator and denominator can only occur when both terms are multiplied as a whole, not simply added.
In this case, the polynomial at the top needs to be converted to the root multiplication that lead to it: (x+1)^2, and the denominator needs to complete the square: (x-1)(x+1)+4, which would still be unable to have terms canceled (as there is still addition in the denominator that cannot be removed), so the original form is the valid answer.
It's a common thing drilled into students during these courses that you cannot simply cancel out terms at will - you have to modify polynomials first.
You can cancel out multipliers that way, but not additions.
In addition to the other great answers, you can really drive the intuition about how wrong this is in students/kids with simple examples:
x+2/x+1-- cancel thexincorrectly and this always equals 2, which should fail the smell check immediately, verify with a couple values ofx.x+1/xcancel thexincorrectly andundefined.Been a while for me too. But the division is probably what breaks it. If X = 3, you get 17/12 vs 7/3.
Around age 12 I read in a recreational maths book that 16/64=16̸/6̸4=1/4 works and I was lucky to encounter this at school while solving a problem on a whiteboard. This is not the case for this fraction but I wonder if there are any non-trivial examples of polynomial division where this works.
Man. I miss when this was the most difficult math thing. Fuck triple integrals.
I fucking hate polynomials.
