### Appendix: Polynomial

##### 1. Root and multiplicity

We have used polynomials as examples of vector spaces. We also use polynomials in the computation of eigenvalue. Here we briefly sketch the basic theory of polynomial.

Polynomials can be defined over any field. A polynomial over a field **F** is

`p`(`t`) = `a`_{0} + `a`_{1}`t` + `a`_{2}`t`^{2} + ... + `a`_{n}t^{n},

where the coefficients `a`_{i} ∈ **F**. Although we will only discuss polynomials with real or complex numbers as coefficients, our discussion will be valid for general fields. Whenever a claim requires complex number, the claim will still be valid for algebraically closed fields.

A root of the polynomial `p`(`t`) is a number `r` satisfying `p`(`r`) = 0. We will show that

`p`(`r`) = 0 ⇔ `p`(`t`) = (`t` - `r`)`q`(`t`) for another polynomial `q`.

By repeatedly using this, we have

`p`(`r`) = 0 ⇔ `p`(`t`) = (`t` - `r`)^{k}s(`t`) for an integer `k` > 0 and another polynomial `s` satisfying `s`(`r`) ≠ 0.

The natural number `k` is the multiplicity of the root `r`. Thus we have the following terminologies

`k` = 1: `r` is a simple root

`k` = 2: `r` is a double root

`k` = 3: `r` is a triple root

`k` > 1: `r` is a multiple root

For example, - `t`^{2} + `t`^{4} = `t`^{2}(`t` + 1) (`t` - 1) has 1 and -1 as simple roots, and has 0 as a double root.

Define the derivative of a polynomial by

`p'`(`t`) = `a`_{1} + 2`a`_{2}`t` + ... + `na`_{n}t^{n-1}.

Then we have (`p` + `q`)`'` = `p'` + `q'`, (`cp`)' = `cp'`, (`pq`)`'` = `p'q` + `pq'`. Thus

`r` is a simple root

⇒ `p`(`t`) = (`t` - `r`)`s`(`t`), `s`(`r`) ≠ 0

⇒ `p'`(`t`) = `s`(`t`) + (`t` - `r`)`s'`(`t`), `s`(`r`) ≠ 0

⇒ `p'`(`r`) = `s`(`r`) + (`r` - `r`)`s'`(`r`) = `s`(`r`) ≠ 0.

On the other hand,

`r` is a multiple root

⇒ `p`(`t`) = (`t` - `r`)^{k}s(`t`), `k `> 1

⇒ `p'`(`t`) = `k`(`t` - `r`)^{k-1}`s`(`t`) + (`t` - `r`)^{k}s'(`t`), `k `> 1

⇒ `p'`(`r`) = `k`(`r` - `r`)^{k-1}`s`(`r`) + (`r` - `r`)^{k}s'(`r`) = 0.

Thus we have the following criterion for multiple roots.

`r` is a multiple root of `p`(`t`) ⇔ `p`(`r`) = 0, `p'`(`r`) = 0.