Why don't all rational equations have a single solution?

Answer 1

The number of solutions of a rational equation depends on the power (or degree) of the equation (that is, the highest power to which the variable is raised) and the domain. In the complex domain, each rational equation of power n has n solutions. It is, however, possible that two or more of these solutions are coincident - or "multiple zeros". In the real domain, the number of solutions can fall in pairs. So an equation of power 7 will always have 7 complex solutions but it can have 7, 5, 3 or 1 real solutions. (Real numbers are a subset of complex numbers).

Another way of seeing this is through factorisation:

The equation x3 + x2 - 10x + 8 = 0 can be factorised into (x + 1)*(x - 2)*(x - 4) = 0

Now the product of three numbers is 0 is any one of them is 0.

That is, if x + 1 = 0 or if x - 2 = 0 or if x - 4 = 0

Thus the equation has the solutions: x = - 1, x = 2 or x = 4.

The equation of order n can have at most n real binomial factors. (Any more and the biggest power of x would be bigger than n). And again, in the complex domain, using the binomial equation (and equivalents), it must have n binomial factors.