Logarithmic Polynomial?
is a polynomial with degree of 5. Given that , , , , and . What is the coefficient of in the polynomial? If the answer can be expressed in the form of , where and are coprime positive integers, find the value of ?
The answer is 283.
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1 solution
Define the quantic polynomials p j ( x ) = 0 ≤ i ≤ 5 , i = j ∏ ( x − 3 i ) 0 ≤ j ≤ 5 Then p j ( 3 i ) = 0 for any 0 ≤ i , j ≤ 5 with i = j , and so the polynomial we want is P ( x ) = j = 1 ∑ 5 p j ( 3 j ) j p j ( x ) = 1 4 2 8 9 8 5 8 1 x 5 − 4 7 2 3 9 2 0 1 2 1 x 4 + 2 5 5 8 7 9 6 0 5 x 3 − 8 7 4 8 6 0 5 x 2 + 1 6 2 1 2 1 x − 1 2 5 8 4 0 8 5 5 8 3 making the answer 1 2 1 + 1 6 2 = 2 8 3 .