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==<span style="font-size:20px; color: blue;">Simple Number Theory</span>==
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==<span style="font-size:20px; color: blue;">Algebra</span>==
This is a collection of essential AIME-level number theory theorems and other tidbits.
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This is a collection of algebra laws and definitions. Obviously, there is '''''WAY''''' too much to cover here, but we'll try to give a good overview.
  
===Trivial Inequality===
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==Elementary Algebra==
For any real <math>x</math>, <math>x^2\ge 0</math>, with equality iff <math>x=0</math>.
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===Definitions===
===Arithmetic Mean/Geometric Mean Inequality===
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*A polynomial is an expression that has exponents that are positive integer constants, and has no higher-level operations or functions.
For any set of real numbers <math>S</math>, <math>\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}</math> with equality iff <math>S_1=S_2=S_3...=S_{k-1}=S_k</math>.
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*A polynomial has degree <math>c</math> if the highest exponent of a variable is <math>c</math>.
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*A quadratic equation is a polynomial of degree <math>2</math>. A quartic is of degree <math>4</math>. A quintic is of degree <math>5</math>.
  
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===Factor Theorem===
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Iff a polynomial <math>P(x)</math> has roots <math>a,b,c,d,e,\ldots,z</math>, then <math>(x-a)(x-b)\ldots (x-z)=0</math>, and <math>(x-a),(x-b)\ldots (x-z)</math> are all factors of <math>P(x)</math>.
  
===Cauchy-Schwarz Inequality===
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===Quadratic Formula===
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For a quadratic of form <math>ax^2+bx+c=0</math>, where <math>a,b,c</math> are constants, the equation has roots <math>\frac{-b\pm\sqrt{b^2-4ac}}{2a}</math>
  
For any real numbers <math>a_1,a_2,...,a_n</math> and <math>b_1,b_2,...,b_n</math>, the following holds:
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===Fundamental Theorems of Algebra===
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*A polynomial of degree <math>n</math> has at least one root, real or complex.
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*A polynomial of degree <math>n</math> has exactly <math>n</math> roots, real or complex.
  
<math>(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2</math>
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===Third-degree and Quartic Formulas===
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If third-degree polynomial <math>Q(x)=ax^3+bx^2+cx+d</math> has roots <math>r,s,t</math>, then:
  
====Cauchy-Schwarz Variation====
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<math>rst=\frac{-b}{a}</math>
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<math>r+s+t=\frac{-c}{a}</math>
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<math>rs+st+rt=\frac{-c}{b}</math>
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<!-- actually, I'm not sure if this is right. Could someone check this? -->
  
For any real numbers <math>a_1,a_2,...,a_n</math> and positive real numbers <math>b_1,b_2,...,b_n</math>, the following holds:
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Quartic formulas are listed [http://www.josechu.com/ecuaciones_polinomicas/cuartica_solucion.htm here].
  
<math>\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}</math>.
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The general quintic equation (or an equation of even higher degree) does not have a formula.
  
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===Determinants===
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The determinant of a <math>2</math> by <math>2</math> (said to have order <math>2</math>) matrix <math>\left |\begin{matrix}a&b \\ c&d\end {matrix}\right|</math> is <math>ad-bc</math>.
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===General Formula for the Determinant===
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Let <math>A</math> be a square matrix of order <math>n</math>. Write <math>A = a_{ij}</math>, where <math>a_{ij}</math> is the entry on the row <math>i</math> and the column <math>j</math>, for <math>i=1,\cdots,n</math> and <math>j=1,\cdots,n</math>. For any <math>i</math> and <math>j</math>, set <math>A_{ij}</math> (called the cofactors) to be the determinant of the square matrix of order <math>n-1)</math> obtained from A by removing the row number i and the column number j multiplied by (-1)^{i+j}. Thus:
  
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<math>\det(A) = \sum_{j=1}^{j=n} a_{ij} A_{ij}</math>
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 +
===Cramer's Law===
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Consider a set of three linear equations (i.e. polynomials of degree one)
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*<math>ax+by+cz=d</math>
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*<math>ex+fy+gz=h</math>
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*<math>ix+jy+kz=l</math>
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Let <math>D=\left|\begin{matrix}a&e&i\\b&f&j\\c&g&k\end{matrix}\right|</math>, <math>D_x=\left|\begin{matrix}d&h&1\\b&f&j\\c&g&k\end{matrix}\right|</math>, <math>D_y=\left|\begin{matrix}a&e&i\\d&h&l\\c&g&k\end{matrix}\right|</math>, <math>D_x=\left|\begin{matrix}a&e&i\\b&f&j\\d&h&l\end{matrix}\right|</math>
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<math>x = \frac{D_x}{D}</math>, <math>y = \frac{D_y}{D}</math>, and <math>z = \frac{D_z}{D}</math>.
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This can be generalized to any number of linear equations.
 +
==Abstract Algebra==
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Incomplete.
 
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Revision as of 12:54, 23 November 2007



The Problem Solver's Resource
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 4.

Algebra

This is a collection of algebra laws and definitions. Obviously, there is WAY too much to cover here, but we'll try to give a good overview.

Elementary Algebra

Definitions

  • A polynomial is an expression that has exponents that are positive integer constants, and has no higher-level operations or functions.
  • A polynomial has degree $c$ if the highest exponent of a variable is $c$.
  • A quadratic equation is a polynomial of degree $2$. A quartic is of degree $4$. A quintic is of degree $5$.

Factor Theorem

Iff a polynomial $P(x)$ has roots $a,b,c,d,e,\ldots,z$, then $(x-a)(x-b)\ldots (x-z)=0$, and $(x-a),(x-b)\ldots (x-z)$ are all factors of $P(x)$.

Quadratic Formula

For a quadratic of form $ax^2+bx+c=0$, where $a,b,c$ are constants, the equation has roots $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Fundamental Theorems of Algebra

  • A polynomial of degree $n$ has at least one root, real or complex.
  • A polynomial of degree $n$ has exactly $n$ roots, real or complex.

Third-degree and Quartic Formulas

If third-degree polynomial $Q(x)=ax^3+bx^2+cx+d$ has roots $r,s,t$, then:

$rst=\frac{-b}{a}$ $r+s+t=\frac{-c}{a}$ $rs+st+rt=\frac{-c}{b}$

Quartic formulas are listed here.

The general quintic equation (or an equation of even higher degree) does not have a formula.

Determinants

The determinant of a $2$ by $2$ (said to have order $2$) matrix $\left |\begin{matrix}a&b \\ c&d\end {matrix}\right|$ is $ad-bc$.

General Formula for the Determinant

Let $A$ be a square matrix of order $n$. Write $A = a_{ij}$, where $a_{ij}$ is the entry on the row $i$ and the column $j$, for $i=1,\cdots,n$ and $j=1,\cdots,n$. For any $i$ and $j$, set $A_{ij}$ (called the cofactors) to be the determinant of the square matrix of order $n-1)$ obtained from A by removing the row number i and the column number j multiplied by (-1)^{i+j}. Thus:

$\det(A) = \sum_{j=1}^{j=n} a_{ij} A_{ij}$

Cramer's Law

Consider a set of three linear equations (i.e. polynomials of degree one)

  • $ax+by+cz=d$
  • $ex+fy+gz=h$
  • $ix+jy+kz=l$

Let $D=\left|\begin{matrix}a&e&i\\b&f&j\\c&g&k\end{matrix}\right|$, $D_x=\left|\begin{matrix}d&h&1\\b&f&j\\c&g&k\end{matrix}\right|$, $D_y=\left|\begin{matrix}a&e&i\\d&h&l\\c&g&k\end{matrix}\right|$, $D_x=\left|\begin{matrix}a&e&i\\b&f&j\\d&h&l\end{matrix}\right|$ $x = \frac{D_x}{D}$, $y = \frac{D_y}{D}$, and $z = \frac{D_z}{D}$. This can be generalized to any number of linear equations.

Abstract Algebra

Incomplete. Back to page 3 | Continue to page 5