# All Odd Degree Polynomials With Real Coefficients Must Have At Least One Real Root One way of coming to this conclusion is to analyze polynomial end behaviors of odd and even degree polynomials (see [[Polynomials#Polynomial End Behavior]]). Because odd degree polynomials have different end behaviors they have to cross the $x$ axis at least once, making the crossing a real root.[^1] [^1]: As per [[The Intermediate Value Theorem]] Another is to remember that [[All Polynomials With Real Coefficients Can Be Factored Into Linear And Irreducible Quadratic Terms]]. If the degree of a polynomial is odd, it means that its factorization must consist of at least one linear term with real coefficients, making this the one linear root. Put another way, **the factorization of a polynomial without any real solutions must consist only of irreducible quadratic terms, and so its degree cannot be odd**.