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Let t be the area of a regular pentagon with each side equal to 1 . Let P(x)=0 be the polynomial equation with least degree, having integer coefficients, satisfied by x=t and the gcd of all the coefficients equal to 1 . If M is the sum of the absolute values of the coefficients of P(x) . What is the integer closest to M ? ( 18^ =( 5 -1) / 2 ) .

Correct answer

16

Step-by-step solution

Area of regular pentagon = a^2 n 4 ( 180 n ) n No. of sides a Length of side For regular pentagon by side length 1 , aligned Area (t) & = 5 4 36^ & = 5 4(0.73) & =1.71 aligned Now, P (1.71)=0 to be found with least degree and integer coefficient soon that gcd of all coefficient is 1 . Let x =1.71100 x=171 P ( x )=100 x -171=0 is the polynomial which satisfied all the conditions. aligned & m =100+171=271 & ~m =16.46 & Nearest integer =16 aligned But this question can have multiple solutions as student can take tan 3

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