Easy polynomial...huh..

Algebra Level 4

consider the equation

( x − 7 ) ( x − 3 ) ( x + 5 ) ( x + 1 ) = 1680 \left( x-7 \right) \left( x-3 \right) \left( x+5 \right) \left( x+1 \right) =1680

On solving for x x , If the number of real roots is N N , the sum of real roots is S S and the product of real roots is P P , find N + S + P N+S+P


The answer is -59.

Krishna Ramesh by Krishna Ramesh , 22, India

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1 solution

Naru Bibi
Apr 28, 2015

Rewrite: ( x − 7 ) ( x + 5 ) ( x − 3 ) ( x + 1 ) = 1680 (x-7)(x+5)(x-3)(x+1) = 1680 = > ( x 2 − 2 x − 35 ) ( x 2 − 2 x − 3 ) = 1680 => (x^{2} - 2x - 35)(x^{2} - 2x - 3) = 1680

For t = x 2 − 2 x − 19 t = x^{2} - 2x - 19 = > t = ( x − 1 ) 2 − 20 > = − 20 => t = (x-1)^{2} - 20 >= -20

Rewrite: ( t − 16 ) ( t + 16 ) = 1680 = > t = 44 (t-16)(t+16) = 1680 => t = 44 (t = -44 is wrong since t >= -20) = > x 2 − 2 x − 63 = 0 = > x = − 7 => x^{2} - 2x - 63 = 0 => x = -7 or x = 9 x = 9 = > N = 2 ; S = 9 − 7 = 2 ; P = 9 ∗ ( − 7 ) = − 63 => N = 2 ; S = 9 - 7 = 2 ; P = 9*(-7) = -63 = > N + S + P = − 59 => N + S + P = -59

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