polynomial equation of odd degree
with odd degree and real coefficients () has at least one real root .
Proof. Denote by the left hand side of (1). We can write
where . But we have because
for all . Thus there exists an such that
since is odd. Therefore the real polynomial function has opposite signs in the end points of the interval . Thus the continuity of guarantees, according to Bolzano’s theorem, at least one zero of in that interval. So (1) has at least one real root .
|Title||polynomial equation of odd degree|
|Date of creation||2013-03-22 15:39:19|
|Last modified on||2013-03-22 15:39:19|
|Last modified by||pahio (2872)|