Odpowiedź:
[tex]a)\\\\3x^5-2x^4-18x^3+12x^2+24x-16=0\\\\3x^5-18x^3+24x-2x^4+12x^2-16=0\\\\3x(x^4-6x^2+8)-2(x^4-6x^2+8)=0\\\\(x^4-6x^2+8)(3x-2)=0\\\\(x^4-2x^2-4x^2+8)(3x-2)=0\\\\(x^2(x^2-2)-4(x^2-2))(3x-2)=0\\\\(x^2-2)(x^2-4)(3x-2)=0\\\\(x-\sqrt{2})(x+\sqrt{2})(x-2)(x+2)(3x-2)=0\\\\\\x-\sqrt{2}=0\ \ \ \ \vee\ \ \ \ x+\sqrt{2}=0\ \ \ \ \vee\ \ \ \ x-2=0\ \ \ \ \vee\ \ \ \ x+2=0\ \ \ \ \vee\ \ \ \ 3x-2=0[/tex]
[tex]x=\sqrt{2}\ \ \ \ \vee\ \ \ \ x=-\sqrt{2}\ \ \ \ \vee\ \ \ \ x=2\ \ \ \ \vee\ \ \ \ x=-2\ \ \ \ \vee\ \ \ \ 3x=2\ \ /:3[/tex]
[tex]\\\\x=\sqrt{2}\ \ \ \ \vee\ \ \ \ x=-\sqrt{2}\ \ \ \ \vee\ \ \ \ x=2\ \ \ \ \vee\ \ \ \ x=-2\ \ \ \ \vee\ \ \ \ x=\frac{2}{3}\\\\\\b)\\\\4x^5-x^4+24x^3-6x^2+20x-5=0\\\\4x^5+24x^3+20x-x^4-6x^2-5=0\\\\4x(x^4+6x^2+5)-(x^4+6x^2+5)=0\\\\(x^4+6x^2+5)(4x-1)=0\\\\(x^4+5x^2+x^2+5)(4x-1)=0\\\\(x^2(x^2+5)+(x^2+5))(4x-1)=0\\\\(x^2+5)(x^2+1)(4x-1)=0\\\\\\x^2+5=0\ \ \ \ \vee\ \ \ \ x^2+1=0\ \ \ \ \vee\ \ \ \ 4x-1=0\\\\x^2=-5\ \ brak\ \ rozwiazania\\\\\\x^2+1=0\\\\x^2=-1\ \ brak\ \ rozwiazania\\\\[/tex]
[tex]4x-1=0\\\\4x=1\ \ /:4\\\\x=\frac{1}{4}\\\\\\c)\\\\2x^4+x^3-4x-2=0\\\\x^3(2x+1)-2(2x+1)=0\\\\(2x+1)(x^3-2)=0\\\\\\2x+1=0\ \ \ \ \ \ \ \ \vee\ \ \ \ x^3-2=0\\\\2x=-1\ \ /:2\ \ \ \ \vee\ \ \ \ x^3=2\\\\x=-\frac{1}{2}\ \ \ \ \ \ \ \ \ \ \ \ \vee\ \ \ \ x=\sqrt[3]{2}[/tex]
[tex]d)\\\\4x^4-5x^2-6=0\\\\4(x^2)^2-5x^2-6=0\\\\Podstawiam\ \ za\ \ x^2=t\\\\4t^2-5t-6=0\\\\4t^2+3t-8t-6=0\\\\t(4t+3)-2(4t+3)=0\\\\(4t+3)(t-2)=0\\\\4t+3=0\ \ \ \ \ \ \ \ \vee\ \ \ \ t-2=0\\\\4t=-3\ \ /:4\ \ \ \ \vee\ \ \ \ t=2\\\\t=-\frac{3}{4}\ \ \ \ \ \ \ \ \ \ \ \vee\ \ \ \ t=2\\\\\\x^2=-\frac{3}{4}\ \ \ \ \vee\ \ \ \ x^2=2\\\\x^2=-\frac{3}{4}\ \ brak\ \ rozwiazania\\\\\\x^2=2\\\\x=\sqrt{2}\ \ \ \ \vee\ \ \ \ x=-\sqrt{2}[/tex]
