Arithmetic Course/Polynominal Equation

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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

$Ax^{2}+Bx+C=0$
$Ax^{2}+C=0$
$Ax^{2}-C=0$

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}}}

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