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Let f(x)=x^2+a x+b . If for all nonzero real xf (x+ 1 x )=f(x)+f ( 1 x ) and the roots of f(x)=0 are integers, what is the value of a^2+b^2 ?

Correct answer

13

Step-by-step solution

aligned & f (x+ 1 x )=f(x)+f ( 1 x ) & (x+ 1 x )^2+a (x+ 1 x )+b=x^2+a x+b+ 1 x^2 + a x +b & b=2 & =2 ( , )=(1,2) or (-1,-2) & a=3 or -3 a^2+b^2=9+4=13 aligned

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