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Let f ( x ) be a polynomial of degree 3 such that f k = - 2 k for k = 2 ,   3 ,   4 ,   5 . Then the value of 52 - 10   f ( 10 ) is equal to _____ .

Correct answer

0

Step-by-step solution

Given that: k f ( k ) + 2 = 0 for k = 2 , 3 , 4 , 5 which means ( k - 2 ) , ( k - 3 ) , ( k - 4 ) , ( k - 5 ) are the factors of this expression. Let k f ( k ) + 2 = a ( k - 2 ) ( k - 3 ) ( k - 4 ) ( k - 5 ) . . . . ( i ) P u t   k = 0 2 = a ( - 2 ) ( - 3 ) ( - 4 ) ( - 5 ) a = 1 60 Put a = 1 60 in ( i ) , we get k f ( k ) + 2 = 1 60 ( k - 2 ) ( k - 3 ) ( k - 4 ) ( k - 5 ) Now, put k = 10 10 f ( 10 ) + 2 = 1 60 × 8 × 7 × 6 × 5 10   f ( 10 ) = 26 So, 52 - 10   f ( 10 ) = 26

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