Cancel out common factors in algebraic expressions that are equal to zero

Question

I have the following original expression:

(100 c)/(e y) + (100 b x)/(e y) + (100 a x^2)/(e y) == 0

which, after factoring out the common terms looks like this intermediate expression:

(100 (c + b x + a x^2))/(e y) == 0

Since 100/(e y) is a common factor, and all of the variables are assumed to be positive numbers, I can cancel it out to obtain the target expression, ready for solving:

c + b x + a x^2 == 0

In summary, I need to be able to "cancel out" all common factors for any kind of algebraic expression, such as in the above example.

Failed Attempt:

Numerator[Factor[(100 a)/(e y) + (100 b x)/(e y) + (100 c x^2)/(e y)]]

gives 100 (a + b x + c x^2) == 0, which does not satisfy the goal of obtaining an expression without common factors.

Answer 1

You need to tell the simplification process your assumptions:

z = FullSimplify[(100 c)/(e y) + (100 b x)/(e y) + (100 a x^2)/(e y) == 0,
             Assumptions -> {e > 0, y > 0}]

c + x (b + a x) == 0

Of course you really don't want to assume that all the variables are positive reals, because then the equation never holds. So we need to either make the assumptions about the variables (like above) or make the assumption about all the variables except the ones that are truly variable (in this case x):

eqn = (100 c)/(e y) + (100 b x)/(e y) + (100 a x^2)/(e y);
var = Variables[eqn];
varNoX = Select[var, # =!= x &];
Factor[FullSimplify[eqn == 0, Assumptions -> Thread[varNoX > 0]]]

c + b x + a x^2 == 0

var contains all the variables, and vars contains all but x, which are assumed to be positive real.