Below is a table of various distances in various units that are important in astronomy and astrophysics.

**Definition:**

The **roots** of a polynomial \(P\) are the solutions of the equation:

\[ P(x) = 0. \]

The product of two linear factors

\[ (x - \alpha)(x - \beta)\]

is a polynomial of degree two:

\[(x - \alpha)(x - \beta) = x^{2} - (\alpha + \beta)x + \alpha\beta.\]

**Definition:**

Polynomials of degree two are also called **quadratic polynomials**.

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**Definition:**

A **polynomial function** of a real variable \(x\), is a function defined by the sum:

\[ P(x) = \sum_{k=0}^{n}c_k x^{k} = c_0 + c_1 x + c_2 x^{2}... + c_n x^{n}. \]

A linear function, \(f(x)\), of a real number, \(x\), is defined by two properties:

- for any real number, \(a\), \(f(ax) = a(f(x));\)
- for any real number, \(h\), \(f(x+h) = f(x) + f(h).\)

For a finite set \(S\) with \(n\) elements, the total number of subsets of \(S\) with \(k\) elements

\[ ( 0 \leq k \leq n) \]

is

\[ \binom{n}{k}.\]

We define \(n\) factorial, written \(n!\), by induction.

For \(n = 1,\)

\[ n! = 1! = 1. \]

For \((n+1)\),

\[ (n+1)! = (n+1)(n!). \]

For a function \(f(n)\) of a natural number \(n,\) we can prove that \(f(n)\) has the property P, for all \(n\), if we can do two things:

- prove that P is true for \(f(n)\) when \(n\) = 1;
- second: prove that
*if*P is true for \(f(n)\),*then*P is also true for \(f(n+1)\).

This process is called mathematical induction.

**Definition:**

For any natural number \(n\), and any natural number \(i\), such that, \( n \geq i \geq 0\):

The **binomial coefficient**,

\[ \binom{n}{i}, \]

of \(n\) and \(i\) is,

\[ \binom {n}{i} := \frac{n!}{(n-i)!(i)!}. \]

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