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
if the highest exponent of a variable is .
- A quadratic equation is a polynomial of degree
. A quartic is of degree . A quintic is of degree .
Factor Theorem
Iff a polynomial has roots , then , and are all factors of .
Quadratic Formula
For a quadratic of form , where are constants, the equation has roots
Fundamental Theorems of Algebra
- A polynomial of degree
has at least one root, real or complex.
- A polynomial of degree
has exactly roots, real or complex.
Third-degree and Quartic Formulas
If third-degree polynomial has roots , then:
The three roots of a cubic polynomial equation are given implicitly by
+ \frac { - 1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c - \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}$$ (Error compiling LaTeX. Unknown error_msg)t=- \frac {a}{3} + \frac { - 1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c + \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3} - \frac {1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c - \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}$
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 by (said to have order ) matrix is .
General Formula for the Determinant
Let be a square matrix of order . Write , where is the entry on the row and the column , for and . For any and , set (called the cofactors) to be the determinant of the square matrix of order obtained from A by removing the row number i and the column number j multiplied by (-1)^{i+j}. Thus:
Cramer's Law
Consider a set of three linear equations (i.e. polynomials of degree one)
Let , , ,
, , and .
This can be generalized to any number of linear equations.
Abstract Algebra
Incomplete.
Diophantine Equations
Incomplete.
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