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

$ax^{2}-b=0$

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

$x=\pm {\sqrt {\frac {b}{a}}}$

$x=\pm {\sqrt {\frac {b}{a}}}$