Storming Polynomial!

Calculus Level 4

If f ( x ) f(x) is a polynomial of degree 6 with leading coefficient 2009 2009 . And that f ( 1 ) = 1 f(1)=1 , f ( 2 ) = 3 f(2)=3 , f ( 3 ) = 5 f(3)=5 , f ( 4 ) = 7 f(4) =7 , f ( 5 ) = 9 f(5)=9 and f ′ ( 2 ) = 2 f'(2)=2 . Find the sum of all the digits of f ( 6 ) f(6) .


The answer is 26.

Hardik Gupta by Hardik Gupta , 22, India

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1 solution

Otto Bretscher
Apr 29, 2016

The given information implies that g ( x ) = f ( x ) − 2 x + 1 = 2009 ( x − 1 ) ( x − 2 ) 2 ( x − 3 ) ( x − 4 ) ( x − 5 ) g(x)=f(x)-2x+1=2009(x-1)(x-2)^2(x-3)(x-4)(x-5) Thus f ( 6 ) = 2009 × 5 ! × 4 + 11 = 964331 f(6)=2009\times5!\times4+11=964331 and the answer is 26 \boxed{26} .

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