zad 1 S = x⁶ - 1 1/4x⁴ + x² - 6 ; T = 1/2x⁴ - x² - 6 ; x = - √2 S - T = x⁶ - 1 1/4x⁴ + x² - 6 - 1/2x⁴ + x² + 6 = x⁶ - 2 3/4x⁴ + 2x² = = (- √2)⁶ - 2 3/4 * (- √2)⁴ + 2 * (- √2)² = 2⁶/² - 11/4 * 2⁴/² + 2 * 2 = 2³ - 11/4 * 2² + 4 = 8 - 11/4 * 4 + 4 = 8 - 11 + 4 = 12 - 11 = 1 zad 2 (x² + 1)(1 - x²) - (x² - α)² = (1 + x²)(1 - x²) - (x⁴ - 2x²α + α²) = = 1 - x⁴ - x⁴ + 2x²α - α² = - 2x⁴ + 2x²α - α² + 1 zad 3 (x - 5)(x² + 5x + 25) ; x = ∛5 x³ - 5x² + 5x² - 25x + 25x - 125 = x³ = (∛5)³ = 5 zad 4 (x² - 4)(x - 1) = 0 (x - 2)(x + 2)(x -1) = 0 x₁ = 2 , x₂ = - 2 , x₃ = 1 x₁ + x₂ + x₃ = 2 - 2 + 1 =₁ 1 zad 5 x(4x - 3)(x - 5) = 0 przedział (- π , π) x = 0 lub 4x - 3 = 0 lub x - 5 = 0 x = 0 lub x = 3/4 lub x = 5 x₁ = 0 i x₂ = 3/4 należą do przedziału zad 6 a) x(x² - 25) = 0 x(x - 5)(x + 5) = 0 x = 0 lub x - 5 = 0 lub x + 5 = 0 x₁ = 0 , x₂ = 5 , x₃ = - 5 b) x(x² - x - 6) = 0 x = 0 lub x² - x - 6 = 0 x² - x - 6 = 0 Δ = (- 1)² - 4 * 1 * (- 6) = 1 + 24 = 25 √Δ = √25 = 5 x₁ = (1 - 5)/2 = - 4/2 = - 2 x₂ = (1 + 5)/2 = 6/2 = 3 x(x + 2)(x - 3) = 0 x = 0 lub x + 2 = 0 lub x -3 = 0 x₁ = 0 , x₂ = - 2 , x₃ = 3 c) 9x⁴ - 6x³ + x² = 0 x²(9x² - 6x + 1) = 0 x² = 0 lub 9x² - 6x + 1 = 0 9x² - 6x + 1 = 0 Δ = (- 6)² - 4 * 9 * 1 = 36 - 36 = 0 x₁ = x₂ = 6/18 = 1/3 x²(x - 1/3)(x - 1/3) = 0 x² = 0 lub x - 1/3 = 0 x₁ = 0 , x₂ = 1/3
