substitution principle

The substitution principle refers to the useful practice of replacing instances of a variable with a different variable. This often simplifies the notation in a way that facilitates finding a desired solution. For instance, the polynomial \(x^6+x^3-6\) cannot easily be factored directly, and there is no general formula for finding the roots of a sixth-degree equation. However, making the substitution \(u=x^3\) in this polynomial yields \(u^2+u-6\), which is trivial to factor as \((u-2)(u+3)\). Substituting \(x^3\) back in for \(u\) gives \((x^3-2)(x^3+3)\), and the roots are thus \(\sqrt[3]{2}\) and \(\sqrt[3]{-3}\).

This technique is especially useful in calculus, where making clever substitutions is sometimes essential in finding a derivative or anti-derivative.

Abstract Algebra

A theorem of commutative ring homomorphisms:

Thm: Given commutative rings \(R\) and \(R^{\prime}\), a homomorphism \(f\) from \(R\) to \(R^{\prime}\), and an element \(r\) of \(R^{\prime}\), there is a unique homomorphism \(F\) from the polynomial ring \(R[x]\) over \(R\) to \(R^{\prime}\) with the properties (1) \(F(a)=f(a)\) for all \(a\) in \(R\), and (2) \(F(x)=r\).

For any polynomial \(p\), \(F_r(p)\) is written \(p(r)\), and corresponds to ‘evaluating \(p\) on \(r\).’ Intuitively, if we ‘plug’ a value into a formal polynomial, we get what we expect.