Zadanie w załączniku ;)
a)
(1 - 2x)³ = 1³ - 3 · 1² · 2x + 3 · 1 · (2x)² - (2x)³ = 1 - 6x + 3 · 4x² - 8x³ = - 8x³ + 12x² - 6x + 1
b)
(5x + 1)³ = (5x)³ + 3 · (5x)² · 1 + 3 · 5x · 1² + 1³ = 125x³ + 3 · 25x² + 15x + 1 = 125x³ + 75x² + 15x + 1
c)
(x - √2)³ = x³ - 3 · x² · √2 + 3 · x · (√2)² - (√2)³ = x³ - 3√2x² + 6x - 2√2
d)
(x + 2)²(x² - 2x + 4) = (x² + 4x + 4)(x² - 2x + 4) = x⁴ - 2x³ + 4x² + 4x³ - 8x² + 16x + 4x² - 8x + 16 = x⁴ + 2x³ + 8x + 16
e)
(3x - 1)²(9x² + 3x + 1) = (9x² - 6x + 1)(9x² + 3x + 1) = 81x⁴ + 27x³ + 9x² - 54 x³ - 18x² - 6x + 9x² + 3x + 1 = 81x⁴ - 27x³ - 3x + 1
f)
(x² + 5x + 25)(x - 5)² = (x² + 5x + 25)(x² - 10x + 25) = x⁴ - 10x³ + 25x² + 5x³ - 50x² + 125x + 25x² - 250x + 625 = x⁴ - 5x³ - 125x + 625
g)
(x + 2)(x² + 4)(x - 2) = (x + 2)(x - 2)(x² + 4) = (x² - 4)(x² + 4) = x⁴ - 16
h)
(x - 1)(x² + x + 1)(x³ + 1) = (x³ - 1)(x³ + 1) = x⁶ - 1
i)
(x + 2)(x⁴ + 4x² + 16)(x - 2) = (x² - 4)(x⁴ + 4x² + 16) = x⁶ - 64
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a)
(1 - 2x)³ = 1³ - 3 · 1² · 2x + 3 · 1 · (2x)² - (2x)³ = 1 - 6x + 3 · 4x² - 8x³ = - 8x³ + 12x² - 6x + 1
b)
(5x + 1)³ = (5x)³ + 3 · (5x)² · 1 + 3 · 5x · 1² + 1³ = 125x³ + 3 · 25x² + 15x + 1 = 125x³ + 75x² + 15x + 1
c)
(x - √2)³ = x³ - 3 · x² · √2 + 3 · x · (√2)² - (√2)³ = x³ - 3√2x² + 6x - 2√2
d)
(x + 2)²(x² - 2x + 4) = (x² + 4x + 4)(x² - 2x + 4) = x⁴ - 2x³ + 4x² + 4x³ - 8x² + 16x + 4x² - 8x + 16 = x⁴ + 2x³ + 8x + 16
e)
(3x - 1)²(9x² + 3x + 1) = (9x² - 6x + 1)(9x² + 3x + 1) = 81x⁴ + 27x³ + 9x² - 54 x³ - 18x² - 6x + 9x² + 3x + 1 = 81x⁴ - 27x³ - 3x + 1
f)
(x² + 5x + 25)(x - 5)² = (x² + 5x + 25)(x² - 10x + 25) = x⁴ - 10x³ + 25x² + 5x³ - 50x² + 125x + 25x² - 250x + 625 = x⁴ - 5x³ - 125x + 625
g)
(x + 2)(x² + 4)(x - 2) = (x + 2)(x - 2)(x² + 4) = (x² - 4)(x² + 4) = x⁴ - 16
h)
(x - 1)(x² + x + 1)(x³ + 1) = (x³ - 1)(x³ + 1) = x⁶ - 1
i)
(x + 2)(x⁴ + 4x² + 16)(x - 2) = (x² - 4)(x⁴ + 4x² + 16) = x⁶ - 64