# 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.