Fundamental Mathematics/Arithmetic/Polynomial Equation

Polynomial Equation

Polynomial Equation has general form

$A_{n}x^{n}+A_{n-1}x^{n-1}+A_{1}x^{1}+A_{o}x^{0}=0$ 1st ordered polynomial equation

1st ordered polynomial equation general form

$Ax+B=0$ Divide the quadratic equation by a, which is allowed because a is non-zero:

$x+{\frac {B}{A}}=0$ $x=-{\frac {B}{A}}$ In summary,

1st ordered polynomial equation $Ax+B=0$ has root $x=-{\frac {B}{A}}$ 2nd ordered polynomial equation

2nd ordered polynomial equation has general form

$Ax^{2}+Bx+C=0$ The quadratic formula can be derived with a simple application of technique of completing the square.Divide the quadratic equation by a, which is allowed because a is non-zero:

$x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0.$ Subtract c/a from both sides of the equation, yielding:

$x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}.$ The quadratic equation is now in a form to which the method of completing the square can be applied. Thus, add a constant to both sides of the equation such that the left hand side becomes a complete square:

$x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+\left({\frac {b}{2a}}\right)^{2},$ which produces:

$\left(x+{\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+{\frac {b^{2}}{4a^{2}}}.$ Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:

$\left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.$ The square has thus been completed. Taking the square root of both sides yields the following equation:

$x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac\ }}{2a}}.$ Isolating x gives the quadratic formula:

$x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.$ Summary

 Polynomial Equation Equation Root 1st ordered equation $ax+b=0$ $x=-{\frac {b}{a}}$ 2nd ordered equation $ax^{2}+bx+c=0$ $x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.$ 