Determine the polynomials p(x) satisfying x*p(x−1)=(x−26)*p(x)

meebuigenzg

meebuigenzg

Answered question

2022-10-06

Determine the polynomials p(x) satisfying x p ( x 1 ) = ( x 26 ) p ( x )

Answer & Explanation

Haylie Campbell

Haylie Campbell

Beginner2022-10-07Added 13 answers

It is easy to see that the polynomial p has roots 0 , 1 , . . . , 25 and therefore
p ( x ) = x ( x 1 ) . . . ( x 25 ) g ( x )
where g ( x ) is also polynomial.
Replacing the original equality we obtain that g ( x 1 ) = g ( x ). Now noting g ( x ) g ( 0 ) = h ( x ) polinomial with degree n, we find that h is roots 0 , 1 , . . . , n ie having more root than degree n. Hence g ( x ) = c constant. Finally
p ( x ) = c x ( x 1 ) . . . ( x 25 ) .

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