Odpowiedź:
1.
2x⁴ +x³ - 18x² - 9x = 0
x³(2x + 1) - 9x(2x + 1) = 0
(2x + 1)(x³ - 9x) = 0
x(2x + 1)(x² - 9) = 0
x(2x +1)(x - 3)(x + 3) = 0
x = 0 ∨ 2x + 1 = 0 ∨ x - 3 = 0 ∨ x + 3 = 0
x = 0 ∨ x = - 1/2 ∨ x = 3 ∨ x = - 3
x₀ = { - 3 , - 1/2 , 0 , 3 }
2.
x⁵ - 2x³ - 2x² + 4 = 0
x³(x² - 2) - 2(x² - 2) = 0
(x² - 2)(x³ - 2) = 0
(x - √2)(x + √2)(x - 2)(x² + 2x + 4) = 0
x - √2 = 0 ∨ x + √2 = 0 ∨ x - 2 = 0 ∨ x² + 2x + 4 = 0
sprawdzamy x² + 2x + 4 = 0
a = 1 , b = 2 , c = 4
Δ = b² - 4ac = 2² - 4 * 1 * 4 = 4 - 16 = - 12
Ponieważ a > 0 i Δ < 0 , więc parabola leży całkowicie nad osia OX i równanie dla x ∈ R przyjmuje wartości większe od 0
x - √2 = 0 ∨ x + √2 = 0 ∨ x - 2 = 0
x = √2 ∨ x = - √2 ∨ x = 2
x₀ = { - √2 , √2 , 2 }
3.
x³ - 21x - 20 = 0
(x - 1)(x² + x - 20) = 0
x - 1 = 0 ∨ x² + x - 20 = 0
x = 1 ∨ x² + x - 20 = 0
x² + x - 20 = 0
a = 1 , b = 1 , c = - 20
Δ = b² - 4ac = 1² - 4 * 1 * (- 20) = 1 + 80 = 81
√Δ = √81 = 9
x₁ = (- b - √Δ)/2a = ( - 1 - 9)/2 = - 10/2 = - 5
x₂ = (- b + √Δ)/2a = (- 1 + 9)/2 = 8/2 = 4
x₀ = { - 5 , 1 , 4 }
4.
2x³ - 3x²- 12x - 7 = 0
(x + 1)(2x² - 5x - 7) = 0
x + 1 = 0 ∨ 2x² - 5x - 7 = 0
x = - 1 ∨ 2x² - 5x - 7 = 0
2x² - 5x - 7 =0
a = 2 , b = - 5 , c = - 7
Δ = b² - 4ac = (- 5)² - 4 * 2 * (- 7) =25 + 56 = 81
√Δ = √81 = 9
x₁ = (- b - √Δ)/2a = (5 - 9)/4 = - 4/4 = - 1
x₂ = (- b + √Δ)/2a = (5 + 9)/4 = 14/4 = 3 2/4 = 3 1/2
x₀ = { - 1 , 3 1/2 }
Odpowiedź:
[tex]1)\\\\2x^4+x^3-18x^2-9x=0\\\\x^3(2x+1)-9x(2x+1)=0\\\\x(2x+1)(x^2-9)=0\\\\x(2x+1)(x-3)(x+3)=0\\\\x=0\ \ \ \ \ \ \ \ \vee\ \ \ \ 2x+1=0\ \ \ \ \vee\ \ \ \ x-3=0\ \ \ \ \vee\ \ \ \ x+3=0\\\\x=0\ \ \ \ \vee\ \ \ \ 2x=-1\ \ /:2\ \ \ \ \vee\ \ \ \ x=3\ \ \ \ \ \vee\ \ \ \ x=-3\\\\x=0\ \ \ \ \vee\ \ \ \ x=-\frac{1}{2}\ \ \ \ \ \ \ \ \ \ \ \ \vee\ \ \ \ x=3\ \ \ \ \vee\ \ \ \ x=-3[/tex]
[tex]2)\\\\x^5-2x^3-2x^2+4=0\\\\x^3(x^2-2)-2(x^2-2)=0\\\\(x^2-2)(x^3-2)=0\\\\(x-\sqrt{2})(x+\sqrt{2})(x^3-2)=0\\\\x-\sqrt{2}=0\ \ \ \ \vee\ \ \ \ x+\sqrt{2}=0\ \ \ \ \vee\ \ \ \ x^3-2=0\\\\x=\sqrt{2}\ \ \ \ \ \ \ \ \vee\ \ \ \ x=-\sqrt{2}\ \ \ \ \ \ \ \ \vee\ \ \ \ x^3=2\\\\x=\sqrt{2}\ \ \ \ \ \ \ \ \vee\ \ \ \ x=-\sqrt{2}\ \ \ \ \ \ \ \ \vee\ \ \ \ x=\sqrt[3]{2}[/tex]
[tex]3)\\\\x^3-21x-20=0\\\\x^3+x^2-x^2-x-20x-20=0\\\\x^2(x+1)-x(x+1)-20(x+1)=0\\\\(x+1)(x^2-x-20)=0\\\\(x+1)(x^2+4x-5x-20)=0\\\\(x+1)(x(x+4)-5(x+4))=0\\\\(x+1)(x+4)(x-5)=0\\\\x+1=0\ \ \ \ \vee\ \ \ \ x+4=0\ \ \ \ \vee\ \ \ \ x-5=0\\\\x=-1\ \ \ \ \ \ \vee\ \ \ \ x=-4\ \ \ \ \ \ \vee\ \ \ \ x=5[/tex]
[tex]4)\\\\2x^3-3x^2-12x-7=0\\\\2x^3+2x^2-5x^2-5x-7x-7=0\\\\2x^2(x+1)-5x(x+1)-7(x+1)=0\\\\(x+1)(2x^2-5x-7)=0\\\\(x+1)(2x^2+2x-7x-7)=0\\\\(x+1)(2x(x+1)-7(x+1))=0\\\\(x+1)(x+1)(2x-7)=0\\\\(x+1)^2(2x-7)=0\\\\(x+1)^2=0\ \ \ \ \vee\ \ \ \ 2x-7=0\\\\x+1=0\ \ \ \ \ \ \vee\ \ \ \ 2x=7\ \ /:2\\\\x=-1\ \ \ \ \ \ \ \ \ \vee\ \ \ \ x=\frac{7}{2}[/tex]
