Đáp án:
$ 2(x-3)(x+1)(x^2 - 2x + 11)$
Giải thích các bước giải:
Ta có
$x^4 + (x-4)^4 - 82 = x^4 - 81 + (x-4)^4 - 1$
$= x^4 - 3^4 + [(x-4)^2]^2 - 1^2$
$= (x^2)^2 - (3^2)^2 + [(x-4)^2 - 1][(x-4)^2 + 1]$
$= (x^2 - 3^2)(x^2 + 3^2) + (x^2 - 4x + 4 - 1)(x^2 - 4x + 4 + 1)$
$= (x-3)(x+3)(x^2 + 9) + (x^2 - 4x + 3)(x^2 - 4x + 5)$
$= (x-3)(x+3)(x^2 + 9) + (x-1)(x-3) (x^2 - 4x + 5)$
$= (x-3)[(x+3)(x^2 + 9) + (x-1)(x^2 - 4x + 5)]$
$= (x-3)(x^3 + 3x^2 + 9x + 27 + x^3 -5x^2 + 9x - 5)$
$= (x-3)(2x^3 -2x^2 + 18x + 22)$
$= (x-3)[(2x^3 + 2x^2) - (4x^2 + 4x) + (22x + 22)]$
$= (x-3)[2x^2(x+1) - 4x(x + 1) + 22(x+1)]$
$= (x-3)(x+1)(2x^2 - 4x + 22)$
$= 2(x-3)(x+1)(x^2 - 2x + 11)$
