A polynomial whose roots form an arithmetic progressionLet f be a fourth degree polynomial whose roots form an arithmetic progression. Prove that f′'s roots also form an arithmetic progression.I didn' t make much progress, I just wrote f ( x ) = a ( x − b − r ) ( x − b − 2 r ) ( x − b − 3 r ) ( x − b − 4 r ) and I tried to differentiate, which obviously doesn't help too much.

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A polynomial whose roots form an arithmetic progressionLet f be a fourth degree polynomial whose roots form an arithmetic progression. Prove that f′&#039;s roots also form an arithmetic progression.I didn&#039; t make much progress, I just wrote   f  (  x  )  =  a  (  x  −  b  −  r  )  (  x  −  b  −  2  r  )  (  x  −  b  −  3  r  )  (  x  −  b  −  4  r  ) and I tried to differentiate, which obviously doesn&#039;t help too much.

luthersavage6lm · Accepted Answer

Step 1Clearly, the problem is unaffected by shifting the polynomial horizontally via   x  ↦  x  −  a. Using this observations, we can WLOG that   f  (  x  )  =  k  (  x  −  3  d  )  (  x  −  d  )  (  x  +  d  )  (  x  +  3  d  ). Since   f  (  x  )  =  f  (  −  x  ), we have       f    ′    (  x  )  =  −      f    ′    (  −  x  ).Step 2Note this implies       f    ′    (  0  )  =  0, and if   α is a root of f′(x) then   −  α is too. So the roots of f′(x) are   0  ,  ±  α for some   α  ∈      R   which is clearly an arithmetic progression.

A polynomial whose roots form an arithmetic progression. Let f be a fourth degree polynomial whose roots form an arithmetic progression. Prove that f′'s roots also form an arithmetic progression.

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