[<< wikibooks] 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.

{\displaystyle 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 = 0Rewrite the equation above

x
+

B
A

=
0

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

x
=
−

B
A

{\displaystyle x=-{\frac {B}{A}}}

== Second Order Equation ==
A second order polynomial equation of one variable x has the general form

A

x

2

+
B
x
+
C
=
0

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

A

x

2

+
C
=
0

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

A

x

2

−
C
=
0

{\displaystyle Ax^{2}-C=0}

=== Solving Equation ===

==== Method 1 ====

A

x

2

+
B
x
+
C
=
0

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

x

2

+

B
A

x
+

C
A

=
0

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

x
=
−
α
±
λ

{\displaystyle x=-\alpha \pm \lambda }
Where

α
=
−

B

2
A

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

β
=
−

C
A

{\displaystyle \beta =-{\frac {C}{A}}}

λ
=

α

2

−

β

2

{\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}}
Depending on the value of

λ

{\displaystyle \lambda }
the equation will have the following root
One Real Root

−
α
=
−

B

2
A

{\displaystyle -\alpha =-{\frac {B}{2A}}}
Two Real Roots

−
α
±
λ

{\displaystyle -\alpha \pm \lambda }

−

B

2
A

±

B

2

−
4
A
C

2
A

{\displaystyle -{\frac {B}{2A}}\pm {\sqrt {\frac {B^{2}-4AC}{2A}}}}
Two Complex Roots

−
α
±
j
λ

{\displaystyle -\alpha \pm j\lambda }

−

B

2
A

±
j

B

2

−
4
A
C

2
A

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

==== Method 2 ====

a

x

2

+
b
=
0

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

x

2

+

b
a

=
0

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

x
=
±

b

a

{\displaystyle x=\pm {\sqrt {{b}{a}}}}

x
=
±
j

b
a

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

==== Method 3 ====

a

x

2

−
b
=
0

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

x

2

−

b
a

=
0

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

x
=
±

b
a

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

x
=
±

b
a

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