Arithmetic Course/Polynominal Equation

Polynomial Equation

An equation is an expression of one variable such that

${\displaystyle f(x)=Ax^{n}+Bx^{(n-1)}+x^{1}+x^{0}=0.}$

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

${\displaystyle x+{\frac {B}{A}}=0}$
${\displaystyle x=-{\frac {B}{A}}}$

Second Order Equation

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

1. ${\displaystyle Ax^{2}+Bx+C=0}$
2. ${\displaystyle Ax^{2}+C=0}$
3. ${\displaystyle Ax^{2}-C=0}$

Solving Equation

Method 1

${\displaystyle Ax^{2}+Bx+C=0}$

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

Where

${\displaystyle \alpha =-{\frac {B}{2A}}}$
${\displaystyle \beta =-{\frac {C}{A}}}$
${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}}$

Depending on the value of ${\displaystyle \lambda }$ the equation will have the following root

One Real Root

${\displaystyle -\alpha =-{\frac {B}{2A}}}$

Two Real Roots

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

Two Complex Roots

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

Method 2

${\displaystyle ax^{2}+b=0}$

${\displaystyle x^{2}+{\frac {b}{a}}=0}$
${\displaystyle x=\pm {\sqrt {-{\frac {b}{a}}}}}$
${\displaystyle x=\pm j{\sqrt {\frac {b}{a}}}}$

Method 3

${\displaystyle ax^{2}-b=0}$

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