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Given a polynomial <math>f(x)</math>, with integer coefficients <math>a_i</math>, all rational roots are in the form <math>\frac{p}{q}</math>, where <math>|p|</math> and <math>|q|</math> are [[coprime]] natural numbers, <math>p|a_0</math>, and <math>q|a_n</math>. | Given a polynomial <math>f(x)</math>, with integer coefficients <math>a_i</math>, all rational roots are in the form <math>\frac{p}{q}</math>, where <math>|p|</math> and <math>|q|</math> are [[coprime]] natural numbers, <math>p|a_0</math>, and <math>q|a_n</math>. | ||
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====Determinants==== | ====Determinants==== |
Revision as of 22:55, 10 January 2009
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 a function of the form
, where , and are real numbers, and are called the coefficients.
- A polynomial has degree if the highest exponent of a variable is . The degree of polynomial is expressed as .
- A quadratic equation is a polynomial of degree . A cubic is 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
- Every polynomial not in the form has at least one root, real or complex.
- A polynomial of degree has exactly roots, real or complex.
Rational Root Theorem
Given a polynomial , with integer coefficients , all rational roots are in the form , where and are coprime natural numbers, , and .
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 by removing the row number and the column number multiplied by . 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.
Newton's Sums
Consider a polynomial of degree , Let have roots . Define the following sums:
The following holds:
Vieta's Sums
Let be a polynomial of degree , so , where the coefficient of is and .
We have: