Let f ( x ) = a x 2 + b x + c be a quadratic polynomial with real coefficients a , b , c . Consider the determinant Δ ( x ) = f ( x ) f ′ ( x ) f ′′ ( x ) f ( x + 1 ) f ′ ( x + 1 ) f ′′ ( x + 1 )

Question

Let f ( x ) = a x 2 + b x + c be a quadratic polynomial with real coefficients a , b , c . Consider the determinant Δ ( x ) = ​ f ( x ) f ′ ( x ) f ′′ ( x ) ​ f ( x + 1 ) f ′ ( x + 1 ) f ′′ ( x + 1 ) ​ f ( x + 2 ) f ′ ( x + 2 ) f ′′ ( x + 2 ) ​ ​ . What is the value of Δ ( x ) ?

TestHub · Accepted Answer

Apply column operations C 2 ​ → C 2 ​ − C 1 ​ and C 3 ​ → C 3 ​ − C 2 ​ . Using f ( x + 1 ) − f ( x ) = a ( 2 x + 1 ) + b , f ( x + 2 ) − f ( x + 1 ) = a ( 2 x + 3 ) + b , f ′ ( x + 1 ) − f ′ ( x ) = 2 a , f ′ ( x + 2 ) − f ′ ( x + 1 ) = 2 a , and f ′′ ( x + 1 ) − f ′′ ( x ) = 0 , f ′′ ( x + 2 ) − f ′′ ( x + 1 ) = 0 , the determinant becomes: Δ ( x ) = ​ f ( x ) f ′ ( x ) f ′′ ( x ) ​ a ( 2 x + 1 ) + b 2 a 0 ​ a ( 2 x + 3 ) + b 2 a 0 ​ ​ Expand along the third row: Δ ( x ) = f ′′ ( x ) ​ a ( 2 x + 1 ) + b 2 a ​ a ( 2 x + 3 ) + b 2 a ​ ​ . Since f ′′ ( x ) = 2 a , we have Δ ( x ) = 2 a [ 2 a ( a ( 2 x + 1 ) + b ) − 2 a ( a ( 2 x + 3 ) + b )] = 2 a [ 2 a ( a ( 2 x + 1 + b − 2 x − 3 − b ))] = 2 a [ 2 a ( a ( − 2 ))] = − 8 a 3 .

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