How can a rational equation have more than one solution?

Answer 1

A rational equation can be multiplied by the least common multiple of its denominators to make it into a polynomial equation. The degree of this polynomial is the highest power (of the variable) that appears in it.

It can be proven that a polynomial of degree n must have n roots in the complex domain. However, there may be fewer roots in the real domain. This is because if the coefficients are real then there may be pairs of complex roots [conjugates] which will not count as real roots. Also, there may be identical roots of multiple order. For example,

x4 - 1 = 0 has 4 complex roots. These are 1, -1, i and -i where i is the imaginary root of -1.

There are only 2 real roots -1 and +1.

x4 = 0 has 4 multiple roots, each of which is 0. Thus x = 0 is a root of multiplicity 4.