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−  Show that if <math>p(x)</math> is a polynomial of degree <math>n</math>, then <math>p(x  1)</math> is a polynomial of the same degree.
 
   
−  Definition of polynomial.
 
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−  :<math>p(x) = \sum_{i=0}^n a_i x^i </math>
 
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−  Binomial theorem.
 
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−  :<math>(a + b)^n = \sum_{i=0}^n {n \choose i} a^{ni} b^i </math>
 
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−  Special case.
 
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−  :<math>(x  1)^n = \sum_{i=0}^n {n \choose i} x^{ni} (1)^i </math>
 
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−  Binomial coefficient simmetry.
 
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−  :<math>{n \choose k} = {n \choose nk} </math>
 
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−  Hence:
 
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−  :<math>(x  1)^n = \sum_{i=0}^n {n \choose i} x^i (1)^i </math>
 
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−  :<math>p(x1) = \sum_{i=0}^n a_i (x  1)^i
 
−  = \sum_{i=0}^n \left[ a_i \left( \sum_{k=0}^n {n \choose k} x^k (1)^k \right) \right]
 
−  = \sum_{i=0}^n \sum_{k=0}^i a_i {n \choose k} x^k (1)^k
 
−  = \sum_{i=0}^n \sum_{k=0}^i a_k {n \choose i} x^i (1)^i.</math>
 
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−  QED.
 
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