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The Problem Solver's Resource

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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

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 | style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 4}}
-==<span style="font-size:20px; color: blue;">Simple Number Theory</span>==
+==<span style="font-size:20px; color: blue;">Algebra</span>==
-This is a collection of essential AIME-level number theory theorems and other tidbits.
+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===
+==Elementary Algebra==
-For any real <math>x</math>, <math>x^2\ge 0</math>, with equality iff <math>x=0</math>.
+===Definitions===
-===Arithmetic Mean/Geometric Mean Inequality===
+*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>.
+*A polynomial has degree <math>c</math> if the highest exponent of a variable is <math>c</math>.
+*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>.
+===Factor Theorem===
+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===
+===Quadratic Formula===
+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:
+===Fundamental Theorems of Algebra===
+*A polynomial of degree <math>n</math> has at least one root, real or complex.
+*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>
+===Third-degree and Quartic Formulas===
+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====
+<math>rst=\frac{-b}{a}</math>
+<math>r+s+t=\frac{-c}{a}</math>
+<math>rs+st+rt=\frac{-c}{b}</math>
+<!-- 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:
+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>.
+The general quintic equation (or an equation of even higher degree) does not have a formula.
+===Determinants===
+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>.
+===General Formula for the Determinant===
+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:
+<math>\det(A) = \sum_{j=1}^{j=n} a_{ij} A_{ij}</math>
+===Cramer's Law===
+Consider a set of three linear equations (i.e. polynomials of degree one)
+*<math>ax+by+cz=d</math>
+*<math>ex+fy+gz=h</math>
+*<math>ix+jy+kz=l</math>
+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>
+<math>x = \frac{D_x}{D}</math>, <math>y = \frac{D_y}{D}</math>, and <math>z = \frac{D_z}{D}</math>.
+This can be generalized to any number of linear equations.
+==Abstract Algebra==
+Incomplete.
 [[User:Temperal/The Problem Solver's Resource3|Back to page 3]] | [[User:Temperal/The Problem Solver's Resource5|Continue to page 5]]
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