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==<span style="font-size:20px; color: blue;">Algebra</span>== | ==<span style="font-size:20px; color: blue;">Algebra</span>== | ||
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. | 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== | + | ===Elementary Algebra=== |
− | ===Definitions=== | + | ====Definitions==== |
− | *A polynomial is | + | *A polynomial is a function of the form |
− | *A polynomial has degree <math>c</math> if the highest exponent of a variable is <math>c</math>. | + | <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 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>. | + | *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>. | ||
− | ===Factor Theorem=== | + | ====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>. | 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=== | + | ====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 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=== | + | ====Fundamental Theorems of Algebra==== |
− | * | + | *Every polynomial not in the form <math>f(x)=c</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. | *A polynomial of degree <math>n</math> has exactly <math>n</math> roots, real or complex. | ||
− | === | + | ====Rational Root Theorem==== |
− | + | 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=== | ||
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>. | 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=== | + | ====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 | + | 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 <math>A</math> by removing the row number <math>i</math> and the column number <math>j</math> multiplied by <math>(-1)^{i+j}</math>. Thus: |
<math>\det(A) = \sum_{j=1}^{j=n} a_{ij} A_{ij}</math> | <math>\det(A) = \sum_{j=1}^{j=n} a_{ij} A_{ij}</math> | ||
− | ===Cramer's Law=== | + | ====Cramer's Law==== |
Consider a set of three linear equations (i.e. polynomials of degree one) | Consider a set of three linear equations (i.e. polynomials of degree one) | ||
*<math>ax+by+cz=d</math> | *<math>ax+by+cz=d</math> | ||
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<math>x = \frac{D_x}{D}</math>, <math>y = \frac{D_y}{D}</math>, and <math>z = \frac{D_z}{D}</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. | This can be generalized to any number of linear equations. | ||
− | == | + | |
− | + | ||
− | == | + | ====Newton's Sums==== |
− | + | Consider a polynomial <math>P(x)</math> of degree <math>n</math>, Let <math>P(x)=0</math> have roots <math>x_1,x_2,\ldots,x_n</math>. Define the following sums: | |
+ | *<math>S_1 = x_1 + x_2 + \cdots + x_n</math> | ||
+ | *<math>S_2 = x_1^2 + x_2^2 + \cdots + x_n^2</math> | ||
+ | *<math>\vdots</math> | ||
+ | *<math>S_k = x_1^k + x_2^k + \cdots + x_n^k</math> | ||
+ | *<math>\vdots</math> | ||
+ | |||
+ | The following holds: | ||
+ | *<math>a_nS_1 + a_{n-1} = 0</math> | ||
+ | *<math>a_nS_2 + a_{n-1}S_1 + 2a_{n-2}=0</math> | ||
+ | *<math>a_nS_3 + a_{n-1}S_2 + a_{n-2}S_1 + 3a_{n-3}=0</math> | ||
+ | *<math>\vdots</math> | ||
+ | ====Vieta's Sums==== | ||
+ | Let <math>P(x)</math> be a polynomial of degree <math>n</math>, so <math>P(x)={a_n}x^n+{a_{n-1}}x^{n-1}+\cdots+{a_1}x+a_0</math>, | ||
+ | where the coefficient of <math>x^{i}</math> is <math>{a}_i</math> and <math>a_n \neq 0</math>. | ||
+ | |||
+ | We have: | ||
+ | <cmath>a_n = a_n</cmath> | ||
+ | <cmath> a_{n-1} = -a_n(r_1+r_2+\cdots+r_n)</cmath> | ||
+ | <cmath> a_{n-2} = a_n(r_1r_2+r_1r_3+\cdots+r_{n-1}r_n)</cmath> | ||
+ | <cmath>\vdots</cmath> | ||
+ | <cmath>a_0 = (-1)^n a_n r_1r_2\cdots r_n</cmath> | ||
+ | |||
[[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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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: