The legendre polynomial P(x) has n real zeros
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The Legendre polynomial Pn with n>0 has n simple roots in (−1,1). Proof by contradiction: Assume Pn has m with 0≤m<n pairwise different zeroes x1,x2,… xm of odd multiplicity in (−1,1), i.e. Pn changes sign in xi, and consider the polynomial Zn(x)=(x−x1)⋅(x−x2)… (x−xm).
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