Arithmetic Course/Polynominal Equation

Polynomial Equation

An equation is an expression of one variable such that

$f(x)=A{x}^n+B{x}^{(n-1)}+x^1+x^0=0.$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. $A{x}^2+Bx+C=0$
2. $A{x}^2+C=0$
3. $A{x}^2-C=0$

Solving Equation

Method 1

$A{x}^2+Bx+C=0$

$x^2+\frac{B}{A}x+\frac{C}{A}=0$

$x=-\alpha \pm \lambda$

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

$a{x}^2+b=0$

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

$x=\pm \sqrt{ba}$

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

Method 3

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

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

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

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