# Arithmetic Course/Polynominal Equation

## Polynomial Equation

An equation is an expression of one variable such that

$f(x)=Ax^{n}+Bx^{(n-1)}+x^{1}+x^{0}=0.$ polynomials used to solve the theory of equations.

## Solving Polynomial Equation

Solving polynomial equations involves finding all the values of variable x that satisfy f(x) = 0.

### First Order Equation

A first order polynomial equation of one variable x has the general form

Ax + B = 0

Rewrite the equation above

$x+{\frac {B}{A}}=0$ $x=-{\frac {B}{A}}$ ## Second Order Equation

A second order polynomial equation of one variable x has the general form

1. $Ax^{2}+Bx+C=0$ 2. $Ax^{2}+C=0$ 3. $Ax^{2}-C=0$ ### Solving Equation

#### Method 1

$Ax^{2}+Bx+C=0$ $x^{2}+{\frac {B}{A}}x+{\frac {C}{A}}=0$ $x=-\alpha \pm \lambda$ Where

$\alpha =-{\frac {B}{2A}}$ $\beta =-{\frac {C}{A}}$ $\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}$ Depending on the value of $\lambda$ the equation will have the following root

One Real Root

$-\alpha =-{\frac {B}{2A}}$ Two Real Roots

$-\alpha \pm \lambda$ $-{\frac {B}{2A}}\pm {\sqrt {\frac {B^{2}-4AC}{2A}}}$ Two Complex Roots

$-\alpha \pm j\lambda$ $-{\frac {B}{2A}}\pm j{\sqrt {\frac {B^{2}-4AC}{2A}}}$ #### Method 2

$ax^{2}+b=0$ $x^{2}+{\frac {b}{a}}=0$ $x=\pm {\sqrt {{b}{a}}}$ $x=\pm j{\sqrt {\frac {b}{a}}}$ #### Method 3

$ax^{2}-b=0$ $x^{2}-{\frac {b}{a}}=0$ $x=\pm {\sqrt {\frac {b}{a}}}$ $x=\pm {\sqrt {\frac {b}{a}}}$ 