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We present six Theorems on the univariate real Polynomial, using which wedevelop a new algorithm for deciding the existence of atleast one real root forunivariate integer Polynomials. Our algorithm outputs that no positive realroot exists, if and only if, the given Polynomial is a factor of a realPolynomial with positive coefficients. Next, we define a transformation thattransforms any instance of 3-SAT into a multivariate real Polynomial withpositive coefficients, if and only if, the instance is not satisfiable.

Another approach to decide on real root existence for univariate Polynomials, and a multivariate extension for 3-SAT