A problem by Morgan Blake

Level pending

The polynomial f ( x ) f(x) is such that f ( x 2 + 1 ) ≡ x 4 + 4 x 2 f(x^{2}+1) \equiv x^{4}+4x^{2} and f ( x 2 − 1 ) ≡ a x 4 + 4 b x 2 + c f(x^{2}-1) \equiv ax^{4}+4bx^{2}+c What is the value of a 2 + b 2 + c 2 a^{2}+b^{2}+c^{2} ?


The answer is 17.

Morgan Blake by Morgan Blake , 25, United Kingdom

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1 solution

Crazy Singh
Jan 2, 2014

f ( x 2 + 1 ) = ( x 2 + 1 − 1 ) ( x 2 + 1 + 3 ) f(x^{2}+1 ) = (x^{2}+1-1) (x^{2}+1+3)

f ( x 2 − 1 ) = ( x 2 − 1 − 1 ) ( x 2 − 1 + 3 ) = x 4 − 4 f(x^{2}-1 )=(x^{2}-1-1) (x^{2}-1+3) = x^{4}-4

a = 1 , b = 0 , c = − 4 a=1, b=0, c=-4

so a 2 + b 2 + c 2 = 17 a^{2}+b^{2}+c^{2}=17

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