Difference between revisions of "User:Temperal/The Problem Solver's Resource4"
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| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 4}} | | style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 4}} | ||
− | ==<span style="font-size:20px; color: blue;"> | + | ==<span style="font-size:20px; color: blue;">Algebra</span>== |
− | This is a collection of | + | 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 <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>. | ||
− | === | + | ===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> | ||
− | + | ===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>( | + | ===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: | ||
− | === | + | <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? --> | ||
− | + | Quartic formulas are listed [http://www.josechu.com/ecuaciones_polinomicas/cuartica_solucion.htm here]. | |
− | + | 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]] | [[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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Revision as of 12:54, 23 November 2007
AlgebraThis 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 AlgebraDefinitions
Factor TheoremIff a polynomial Quadratic FormulaFor a quadratic of form Fundamental Theorems of Algebra
Third-degree and Quartic FormulasIf third-degree polynomial
Quartic formulas are listed here. The general quintic equation (or an equation of even higher degree) does not have a formula. DeterminantsThe determinant of a General Formula for the DeterminantLet
Cramer's LawConsider a set of three linear equations (i.e. polynomials of degree one) Let Abstract AlgebraIncomplete. Back to page 3 | Continue to page 5 |