Don't overthink

Algebra Level 3

When x 4 + 2 x^{4}+2 is divided by ( x 2 ) 2 (x-2)^{2} , the remainder can be expressed in the form a x + b ax+b . Find a b a-b .


The answer is 78.

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1 solution

Chaitanya Rao
Mar 4, 2020

By long division, x 4 + 2 = ( x 2 4 x + 4 ) ( x 2 + 4 x + 12 ) + 32 x 46 = ( x 2 ) 2 ( x 2 + 4 x + 12 ) + 32 x 46 x^4 + 2 = (x^2 - 4x + 4)(x^2 + 4x + 12) + 32x - 46 = (x-2)^2(x^2 + 4x + 12) + 32x - 46 , so the answer is 32 + 46 = 78 32 + 46 = \boxed{78} .

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