[<< 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}}}}