Constant term

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a constant term (sometimes referred to as a free term) is a <a href="/facts/Term_(logic)/M0nXp7dH">term</a> in an <a href="/facts/Algebraic_expression/CK7D57Dm">algebraic expression</a> that does not contain any <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a> and therefore is <a href="/facts/Constant_function/BmPx6dn7">constant</a>. For example, in the <a href="/facts/Quadratic_polynomial/mKxHIC3U">quadratic polynomial</a>,

x
          
            2
          
        
        +
        2
        x
        +
        3
        ,
         
      
    
    {\displaystyle x^{2}+2x+3,\ }

The number 3 is a constant term.
After <a href="/facts/Like_terms/fsPC7rXn">like terms</a> are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial

a
        
          x
          
            2
          
        
        +
        b
        x
        +
        c
        ,
         
      
    
    {\displaystyle ax^{2}+bx+c,\ }

where 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is the variable, as having a constant term of 
 
 
 
 c
 .
 
 
 {\displaystyle c.}
 
 If the constant term is 0, then it will conventionally be omitted when the quadratic is written out.
Any <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> written in standard form has a unique constant term, which can be considered a <a href="/facts/Coefficient/5Hwq2Wuq">coefficient</a> of 
 
 
 
 
 x
 
 0
 
 
 .
 
 
 {\displaystyle x^{0}.}
 
 In particular, the constant term will always be the lowest <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial

x
          
            2
          
        
        +
        2
        x
        y
        +
        
          y
          
            2
          
        
        −
        2
        x
        +
        2
        y
        −
        4
         
      
    
    {\displaystyle x^{2}+2xy+y^{2}-2x+2y-4\ }

has a constant term of −4, which can be considered to be the coefficient of 
 
 
 
 
 x
 
 0
 
 
 
 y
 
 0
 
 
 ,
 
 
 {\displaystyle x^{0}y^{0},}
 
 where the variables are eliminated by being exponentiated to 0 (any non-zero number exponentiated to 0 becomes 1). For any polynomial, the constant term can be obtained by substituting in 0 instead of each variable; thus, eliminating each variable. The concept of exponentiation to 0 can be applied to <a href="/facts/Power_series/vYh6sTps">power series</a> and other types of series, for example in this power series:

a
          
            0
          
        
        +
        
          a
          
            1
          
        
        x
        +
        
          a
          
            2
          
        
        
          x
          
            2
          
        
        +
        
          a
          
            3
          
        
        
          x
          
            3
          
        
        +
        ⋯
        ,
      
    
    {\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots ,}

a
 
 0
 
 
 
 
 {\displaystyle a_{0}}
 
 is the constant term.

Constant term open-in-new

Constant term