Упростите выражение (х-2)²(x + 2) + 2(x + 2)(x - 2) и найдите его значение при х = −2.

Решение:

(х-2)²(x + 2) + 2(x + 2)(x - 2) = (x - 2)(x - 2)(x + 2) + 2(x + 2)(x - 2) = (x - 2)(x² - 4) + 2(x² - 4) = x³ - 4x - 2x² + 8 + 2x² - 8 = x³ - 4x

При x = -2: (-2)³ - 4(-2) = -8 + 8 = 0

(х-2)²(x + 2) + 2(x + 2)(x - 2) = x³ - 4x + 2x^2 - 8 = x^3 + 2x^2 -4x -8

При x = -2: (-2)³ + 2(-2)^2 - 4(-2) - 8 = -8 + 8 + 8 - 8 = -16

(х-2)²(x + 2) + 2(x + 2)(x - 2) = (x² - 4x + 4)(x+2) + 2(x²-4) = x^3 + 2x^2 -4x^2 - 8x + 4x + 8 + 2x^2 - 8 = x^3 -4x = (-2)^3 - 4(-2) = -8 + 8 = 0

Раскрываем скобки:

(x-2)²(x+2) + 2(x+2)(x-2) = (x² - 4x + 4)(x+2) + 2(x² - 4) = x³ + 2x² - 4x² - 8x + 4x + 8 + 2x² - 8 = x³ - 4x

Подставляем x = -2:

(-2)³ - 4(-2) = -8 + 8 = 0