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====Definitions==== | ====Definitions==== | ||
*A polynomial is a function of the form | *A polynomial is a function of the form | ||
− | <cmath>f(x)=a_nx^n+a_{n-1}x^{n-1}\ldots+a_0</cmath>, where <math>a_n\ne 0</math>, and <math>a_i</math> are real numbers, and are called the [[coefficients]]. | + | <cmath>f(x)=a_nx^n+a_{n-1}x^{n-1}\ldots+a_0</cmath>, where <math>a_n\ne 0</math>, and <math>a_i</math> are real numbers, and are called the [[Coefficient|coefficients]]. |
*A polynomial has degree <math>c</math> if the highest exponent of a variable is <math>c</math>. The degree of polynomial <math>P</math> is expressed as <math>\deg(P)</math>. | *A polynomial has degree <math>c</math> if the highest exponent of a variable is <math>c</math>. The degree of polynomial <math>P</math> is expressed as <math>\deg(P)</math>. | ||
*A quadratic equation is a polynomial of degree <math>2</math>. A cubic is of degree <math>3</math>. A quartic is of degree <math>4</math>. A quintic is of degree <math>5</math>. | *A quadratic equation is a polynomial of degree <math>2</math>. A cubic is of degree <math>3</math>. A quartic is of degree <math>4</math>. A quintic is of degree <math>5</math>. | ||
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<cmath>\vdots</cmath> | <cmath>\vdots</cmath> | ||
<cmath>a_0 = (-1)^n a_n r_1r_2\cdots r_n</cmath> | <cmath>a_0 = (-1)^n a_n r_1r_2\cdots r_n</cmath> | ||
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[[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]] |
Latest revision as of 21:45, 27 February 2020
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: