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.
Roots of a polynomial - multiplying two terms of (x+some constant)(x+some constant) should equal the equation with a primary term of x^2
“Completing the square” - Attempting to find roots of a difficult polynomial (in the case of that equation, finding 2 easy roots and adding a constant at the end of the denominator)
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.
I did not understand a single word of that but thank you
Numerator - top half of a fraction
Denominator - bottom half of a fraction
Roots of a polynomial - multiplying two terms of (x+some constant)(x+some constant) should equal the equation with a primary term of x^2
“Completing the square” - Attempting to find roots of a difficult polynomial (in the case of that equation, finding 2 easy roots and adding a constant at the end of the denominator)
Also Jia tan lol, XZ utils backdoor username?
Thanks
Shhhhhh I’m on the run. The feds won’t look on blahaj.zone because they’re afraid of DEI accusations.
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.