Does a polynomial of degree n necessarily have n real solutions? Does it necessarily have n unique solutions, real or imaginary?

1 Answer
Oct 24, 2015

Answer:

No. A polynomial equation in one variable of degree #n# has exactly #n# Complex roots, some of which may be Real, but some may be repeated roots.

Explanation:

For example, #0 = x^4+2x^2+1 = (x-i)^2(x+i)^2# has roots #i#, #i#, #-i#, #-i#.

If the polynomial has Real coefficients, then any Complex roots occur as conjugate pairs. So if a polynomial of degree #n# has Real coefficients, then it has #n - 2k# Real roots and #2k# non-Real Complex roots for some integer #k >= 0# counting repeated roots.