Roots of functions / polynomials

It is often very important for us to know when two functions are equal; i.e. to solve an equation $f(x) = g(x)$.

It turns out to greatly simplify the problem if you rewrite the question as $f(x) - g(x) = 0$; so if we learn how to solve equations $h(x) = 0$, that also lets us solve equations $f(x) = g(x)$ too.

As an example of simplification, imagine if you had to learn the quadratic formula as

To solve $a x^2 + b x + c = d x^2 + e x + f$ when $a \neq d$, the roots are $$ x = \frac{(e-b) \pm \sqrt{ (e-b)^2 + 4 (a-d)(f-c) }}{2(a-d)} $$

Gah, much worse!

There's a strong analogy between factoring integers into a product of primes, and factoring polynomials into a product of irreducible polynomials. Since the irreducible factors correspond to roots of the polynomial, this shows that finding roots of a polynomial is like finding the prime factors of an integer. Similarly, finding the roots and poles of a rational function is like finding the prime factorization (allowing negative exponents) of a rational number.

(the roots may be in an extension; e.g. the roots of the irreducible real polynomial $x^2 + 1$ are complex numbers)

The general properties of analytic functions -- roughly, functions equal to their Taylor series -- share a lot of properties with polynomials. Techniques for working with polynomials usually get generalized to work with analytic functions... such as finding the zeroes and poles.

The analysis zeta function is a whole subject in of itself.

For a polynomial $p(x)$ of degree $n$, if we know all the $n$ (complex) roots $r_1, r_2, r_3, ..., r_n$, where some may be equal, we know exactly what the polynomial is:

$$p(x) = C(x-r_1)\cdot(x-r_2)\cdot(x-r_3)\cdot...\cdot(x-r_n)$$

So, in a sense, all the information about the polynomial is contained "within" the roots of the polynomial.