Difference between revisions of "User:Temperal/The Problem Solver's Resource4"

Latest revision as of 20:45, 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

$f(x)=a_nx^n+a_{n-1}x^{n-1}\ldots+a_0$ , where $a_n\ne 0$ , and $a_i$ are real numbers, and are called the coefficients.

A polynomial has degree $c$ if the highest exponent of a variable is $c$ . The degree of polynomial $P$ is expressed as $\deg(P)$ .
A quadratic equation is a polynomial of degree $2$ . A cubic is of degree $3$ . A quartic is of degree $4$ . A quintic is of degree $5$ .

Factor Theorem

Iff a polynomial $P(x)$ has roots $a,b,c,d,e,\ldots,z$ , then $(x-a)(x-b)\ldots (x-z)=0$ , and $(x-a),(x-b)\ldots (x-z)$ are all factors of $P(x)$ .

Quadratic Formula

For a quadratic of form $ax^2+bx+c=0$ , where $a,b,c$ are constants, the equation has roots $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Fundamental Theorems of Algebra

Every polynomial not in the form $f(x)=c$ has at least one root, real or complex.
A polynomial of degree $n$ has exactly $n$ roots, real or complex.

Rational Root Theorem

Given a polynomial $f(x)$ , with integer coefficients $a_i$ , all rational roots are in the form $\frac{p}{q}$ , where $|p|$ and $|q|$ are coprime natural numbers, $p|a_0$ , and $q|a_n$ .

Determinants

The determinant of a $2$ by $2$ (said to have order $2$ ) matrix $\left |\begin{matrix}a&b \\ c&d\end {matrix}\right|$ is $ad-bc$ .

General Formula for the Determinant

Let $A$ be a square matrix of order $n$ . Write $A = a_{ij}$ , where $a_{ij}$ is the entry on the row $i$ and the column $j$ , for $i=1,\cdots,n$ and $j=1,\cdots,n$ . For any $i$ and $j$ , set $A_{ij}$ (called the cofactors) to be the determinant of the square matrix of order $n-1$ obtained from $A$ by removing the row number $i$ and the column number $j$ multiplied by $(-1)^{i+j}$ . Thus:

$\det(A) = \sum_{j=1}^{j=n} a_{ij} A_{ij}$

Cramer's Law

Consider a set of three linear equations (i.e. polynomials of degree one)

$ax+by+cz=d$
$ex+fy+gz=h$
$ix+jy+kz=l$

Let $D=\left|\begin{matrix}a&e&i\\b&f&j\\c&g&k\end{matrix}\right|$ , $D_x=\left|\begin{matrix}d&h&1\\b&f&j\\c&g&k\end{matrix}\right|$ , $D_y=\left|\begin{matrix}a&e&i\\d&h&l\\c&g&k\end{matrix}\right|$ , $D_x=\left|\begin{matrix}a&e&i\\b&f&j\\d&h&l\end{matrix}\right|$ $x = \frac{D_x}{D}$ , $y = \frac{D_y}{D}$ , and $z = \frac{D_z}{D}$ . This can be generalized to any number of linear equations.

Newton's Sums

Consider a polynomial $P(x)$ of degree $n$ , Let $P(x)=0$ have roots $x_1,x_2,\ldots,x_n$ . Define the following sums:

$S_1 = x_1 + x_2 + \cdots + x_n$
$S_2 = x_1^2 + x_2^2 + \cdots + x_n^2$
$\vdots$
$S_k = x_1^k + x_2^k + \cdots + x_n^k$
$\vdots$

The following holds:

$a_nS_1 + a_{n-1} = 0$
$a_nS_2 + a_{n-1}S_1 + 2a_{n-2}=0$
$a_nS_3 + a_{n-1}S_2 + a_{n-2}S_1 + 3a_{n-3}=0$
$\vdots$

Vieta's Sums

Let $P(x)$ be a polynomial of degree $n$ , so $P(x)={a_n}x^n+{a_{n-1}}x^{n-1}+\cdots+{a_1}x+a_0$ , where the coefficient of $x^{i}$ is ${a}_i$ and $a_n \neq 0$ .

We have: $a_n = a_n$ $a_{n-1} = -a_n(r_1+r_2+\cdots+r_n)$ $a_{n-2} = a_n(r_1r_2+r_1r_3+\cdots+r_{n-1}r_n)$ $\vdots$ $a_0 = (-1)^n a_n r_1r_2\cdots r_n$

Back to page 3 | Continue to page 5

@@ Line 1: / Line 1: @@
 __NOTOC__
-<br /><br />
+{{User:Temperal/testtemplate|page 4}}
-{| style='background:lime;border-width: 5px;border-color: limegreen;border-style: outset;opacity: 0.8;width:840px;height:300px;position:relative;top:10px;'
-|+ <span style="background:aqua; border:1px solid black; opacity: 0.6;font-size:30px;position:relative;bottom:8px;border-width: 5px;border-color:blue;border-style: groove;position:absolute;top:50px;right:155px;width:820px;height:40px;padding:5px;">The Problem Solver's Resource</span>
-|-
-| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 4}}
 ==<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.
-==Elementary Algebra==
+===Elementary Algebra===
-===Definitions===
+====Definitions====
-*A polynomial is an expression that has exponents that are positive integer constants, and has no higher-level operations or functions.
+*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>.
-===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>
-===Fundamental Theorems of Algebra===
+====Fundamental Theorems of Algebra====
-*A polynomial of degree <math>n</math> has at least one root, real or complex.
+*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.
-===Third-degree and Quartic Formulas===
+====Rational Root Theorem====
-If third-degree polynomial <math>Q(x)=ax^3+bx^2+cx+d</math> has roots <math>r,s,t</math>, then:
+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>.
-*<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====
-===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===
+====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:
+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>
-===Cramer's Law===
+====Cramer's Law====
 Consider a set of three linear equations (i.e. polynomials of degree one)
 *<math>ax+by+cz=d</math>
@@ Line 51: / Line 41: @@
 <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.
-==Diophantine Equations==
+====Newton's Sums====
-Incomplete.
+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]]
-|}<br /><br />

Page

Toolbox

Search