The purpose of this paper is to apply ultraproduct techniques to some problems in infinite dimensional holomorphy. The central problem we consider is the following: Given a continuous polynomial \(P\), or more generally, a holomorphic function, \(f\), defined on a Banach space \(X\), can we extend \(P\) or \(f\) to a larger space containing \(X\)? Questions such as these were first tackled by \textit{R. M. Aron} and \textit{P. D. Berner} [Bull. Soc. Math. France 106, 3-24 (1978; Zbl 0378.46043)]. They showed how polynomials, and certain holomorphic functions, can be extended to the bidual \(X^{**}\). From this, they were able to construct extensions for other spaces containing \(X\). However, some questions were left open. For example, it was not known whether the Aron-Berner extension of a continuous polynomial \(P\) from \(X\) to \(X^{**}\) had the same norm as \(P\). This question was recently answered in the affirmative by \textit{A. M. Davie} and \textit{T. W. Gamelin} [Proc. Am. Math. Soc. 106, No. 2, 351-356 (1989; Zbl 0683.46037)]. We present a new approach to this extension problem. Our approach is to work with an ultrapower \((X)_u\) of the Banach space \(X\) rather than the bidual of \(X\). There is a canonical embedding of \(X\) into \((X)_u\), and it is relatively simple to construct extensions of polynomials and holomorphic functions from \(X\) into \((X)_u\). For certain special ultrapowers of \(X\) we have roughly speaking, \(X\subset X^{**} \subset (X)_u\), and so we obtain extensions from \(X\) to its bidual as byproduct of our extension process. There is not one, but several ultrapower extension processes. One of these processes is modelled on the Aron-Berner method, and in this case we extend the scope of the result of Davie and Gamelin mentioned above. The other extension process which we discuss is more adaptable for dealing with holomorphic functions. Our methods yield new results concerning the polarization constants of a Banach space. The polarization constants of \(X\) are a sequence of real numbers \(K_n (X)\) which contain information about the geometric structure of \(X\). The number \(K_n (X)\) arises when one compares the norm of a homogeneous polynomial of degree \(n\) on \(X\) with the norm of the symmetric \(n\)-linear function which generates the polynomial. We show that the bidual \(X^{**}\) has the same polarization constants as \(X\), at least when the bidual has the metric approximation property.