6th degree polynomial?

Algebra Level 4

Find the sum of all distinct real solutions to the equation ( x 2 6 x + 8 ) 3 + ( x 2 3 x 4 ) 3 = ( 2 x 2 9 x + 4 ) 3 (x^2-6x+8)^3+(x^2-3x-4)^3=(2x^2-9x+4)^3 .


The answer is 5.5.

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1 solution

Renz Mina
Aug 2, 2016

Let u = x 2 6 x + 8 u=x^2-6x+8 and v = x 2 3 x 4 v=x^2-3x-4 . Notice that u 3 + v 3 = ( u + v ) 3 u^3+v^3=(u+v)^3 . Simplifying this, we have u v ( u + v ) = 0 uv(u+v)=0 . That is ( x 2 6 x + 8 ) ( x 2 3 x 4 ) ( 2 x 2 9 x + 4 ) = 0 (x^2-6x+8)(x^2-3x-4)(2x^2-9x+4)=0 . We get x = 2 , 4 , 1 , 0.5 x=2,4,-1,0.5 Hence the answer is 5.5 5.5 .

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