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User:Temperal/The Problem Solver's Resource4

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< User:Temperal

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)

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:

The following holds:

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

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