Fundamentals of Algebra - Cancelling Like Terms in Divison

Question

When can, or can't, terms be cancelled.

ie: $\frac{3x^2-1}{x^2}$

$x^2$ cannot be cancelled. Why not, and what are the rules?

Answer 1

1. Anything trig over/above something not trig.

1.a. unlike trig functions.

1. Anytime there exists a higher power of a variable above or below some smaller power.
2. Anytime there is a constant above or below with addition or subtraction operators with variables.

And of course this is all given that a series of variables with powers doesn't equal the below or above.

Answer 2

Since “canceling” means “dividing the numerator and the denominator by the same (non-zero) number” one might cancel the given fraction by $7$, e.g., which yields to $\dfrac{\dfrac{3x^2}{7}-\dfrac{1}{7}}{\dfrac{x^2}{7}},$ which is perfectly right and perfectly senseless either.

Moral: canceling means not simplifying per se. If simplifying is the goal, first factorize numerator and denominator and look for common factors.