Odpowiedź:
a)
6x⁴ + 2x³ = 0
2x³(3x + 1) = 0
2x³ = 0 ∨ 3x + 1 = 0
x = 0 ∨ 3x = - 1
x = 0 ∨ x = - 1/3
b)
- x⁵ + 4x³ = 0
- x³(x² - 4) = 0
- x³ = 0 ∨ x² - 4 = 0
x = 0 ∨ (x - 2)(x + 2) = 0
x = 0 ∨ x = 2 ∨ x = - 2
c)
x⁶ + x = x + 2x⁴
x⁶ - 2x⁴ - x - x = 0
x⁶ - 2x⁴ = 0
x⁴(x² - 2) = 0
x⁴ = 0 ∨ x² - 2 = 0
x = 0 ∨ (x - √2)(x + √2) = 0
x = 0 ∨ x - √2 = 0 ∨ x + √2 = 0
x = 0 ∨ x = √2 ∨ x = - √2
d)
(x⁴ - 4x³)/2 = (x⁵ - 6x³)/3
3(x⁴ - 4x³) = 2(x⁵ - 6x³)
3x⁴ - 12x³ = 2x⁵ - 6x³
- 2x⁵ + 3x⁴ - 12x³ + 6x³ = 0
- 2x⁵ + 3x⁴ - 6x³ = 0
x³(- 2x² + 3x - 6) = 0
x³ = 0 ∨ - 2x² + 3x - 6 = 0
x = 0
- 2x² + 3x - 6 = 0
a = - 2 , b = 3 , c = - 6
Δ = b² - 4ac = 3² - 4 * (- 2) * ( - 6) = 9 - 48 = - 39
Δ < 0 i a < 0 ; równanie kwadratowe przyjmuje dla x ∈ R tylko wartości mniejsze od 0 , więc :
x = 0
e)
1/6x³ + 2x² + 6x = 0 | * 6
x³ + 12x² + 36x = 0
x(x² + 12x + 36) = 0
x = 0 ∨ x² + 12x + 36 = 0
x² + 12x + 36 = 0
a = 1 , b = 12 , c = 36
Δ = b² - 4ac = 12² - 4 * 1 * 36 = 144 - 144 = 0
x₁ = x₂ = - b/2a = - 12/2 = - 6
x = 0 ∨ x = - 6
f)
- 4x⁴ + 4x³ - x² = 0
- x²(4x² - 4x + 1) = 0
- x² = 0 ∨ 4x² - 4x + 1 = 0
x = 0
x² - 4x + 1 = 0
a = 1 , b = - 4 , c = 1
Δ = b² - 4ac = (- 4)² - 4 * 1 * 1 = 16 - 16 = 0
x₁ = x₂ = - b/2a = 4/2 = 2
x = 0 ∨ x = 2
∨ - znaczy "lub"
