Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

CERTAIN GENERAL METHODS

173

4. The Equation uv= 0. Consider, for example, the equation
(1) X2 - y2 = 0.
Since X2 - y2 ( - y)( + y),
it is clear that equation (1) will be satisfied
(a) if (x, y) lies on the line
(2) x-y=0;
(b) if (x, y) lies on the line
(3) x+y= 0;
and in no other case. Equation (1), therefore, is equivalent to
the two equations (2) and (3) taken together, and it represents,
therefore, the two right lines (2) and (3).
It is clear from this example that we can generalize and
say:
THEOREM. The equation
uv = 0
represents those points (x, y) which lie on each of the two curves,
u = O, v = 0,
and no others.
It follows as an immediate consequence of the theorem that
the equation w... =0,
whose left-hand member is the product of any number of
factors, represents the totality of curves corresponding to the
individual factors, when these are successively set equal to zero.
Example. Consider the equation,
4 - y4 = 0.
Here,x
4 - y4 = (x2 - y2) (2 + y2) = ( - y) (x + y) (X2 + y2).
* It is true that the following equation is an identity, and so the sign
instead of = might be expected. The use of the sign _ for an identical
equation is not, however, considered obligatory, the sign = being used
when it is clear that the equation is an identity, so that the fact does
not require special emphasis.