Bernstein polynomials.

    The Bernstein polynomials of degree are  
    
, for ,  where .  

Exploration.

Graphs of the Bernstein polynomials of degree  .

Observations.  

(i)    The Bernstein polynomials are non-negative over the interval .  

(ii)    ,  which we can easily verify for the case ;

Property (ii) is merely the simplification    

(iii)    The Bernstein polynomials of degree form a basis of the subspace of continuous functions spanned by .  
    For our purposes we only need this fact for degree .  
    
    That is, the span of   and the span of are the same.  It will suffice to show that each of the functions is a linear combination of the Bernstein polynomials.  Observation (ii) shows that 1 is a linear combination, and by definition is in both set.   We need to check out .   We find that   
    
    ,    
and  
    .  

(c) John H. Mathews 2003