# MandysNotes

Lessons

### Table of Distances

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

Sunday, 09 March 2014 16:59

### The Roots of a Polynomial

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Definition:

The roots of a polynomial $$P$$ are the solutions of the equation:

$P(x) = 0.$

Sunday, 09 March 2014 16:51

### Polynomials of Degree Two

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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.

Saturday, 08 March 2014 03:01

### The Definition of a Polynomial in One Real Variable

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Definition:

A  polynomial function of a real variable $$x$$, is a function defined by the sum:
$P(x) = \sum_{k=0}^{n}c_k x^{k} = c_0 + c_1 x + c_2 x^{2}... + c_n x^{n}.$

Saturday, 08 March 2014 01:36

### Linear Functions and Linear Equations

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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, $$h$$, $$f(x+h) = f(x) + f(h).$$
Saturday, 08 March 2014 01:31

### Binomial Coefficients and Subsets

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For a finite set  $$S$$ with $$n$$ elements, the total number of subsets of $$S$$ with $$k$$ elements

$( 0 \leq k \leq n)$

is

$\binom{n}{k}.$

Friday, 07 March 2014 21:11

### The Factorial

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We define $$n$$ factorial, written $$n!$$, by induction.

For  $$n = 1,$$
$n! = 1! = 1.$
For $$(n+1)$$,
$(n+1)! = (n+1)(n!).$

Friday, 07 March 2014 21:08

### Mathematical Induction

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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.

Friday, 07 March 2014 21:05

### Binomial Coefficients

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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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http://latex-project.org/ftp.html

This site has valuable information:

Tex-Latex Stack Exchange:

http://tex.stackexchange.com/

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