Zadania w załączniku! Z góry dziękuję

[tex]zad.1\\\\x^{5} -2x^{3} =0\\\\x^{3} \cdot (x^{2} -2)=0\\\\x^{3} \cdot (x^{2} -(\sqrt{2} )^{2} )=0\\\\x^{3} \cdot (x-\sqrt{2} )\cdot (x+\sqrt{2} )=0\\\\x^{3} =0~~\lor~~x-\sqrt{2} =0~~\lor ~~(x+\sqrt{2} )=0\\\\x=0~~\lor ~~x=\sqrt{2} =0~~\lor~~x=-\sqrt{2}[/tex]

[tex]zad.2\\\\W(x)=2x^{3} -3x^{2} -6x+9\\\\W(x)=2x^{3} -6x-3x^{2} +9\\\\W(x)=2x(x^{2} -3)-3(x^{2} -3)\\\\W(x)=(x^{2} -3)\cdot (2x-3)\\\\W(x)=(x^{2} -(\sqrt{3} )^{2} )\cdot (2x-3)\\\\W(x)=(x-\sqrt{3} )\cdot (x+\sqrt{3} )\cdot (2x-3)[/tex]

Jeśli szukamy pierwiastków wielomainu W(x) czyli szukamy rozwiązań równania W(x)=0

[tex]W(x)=(x-\sqrt{3} )\cdot (x+\sqrt{3} )\cdot (2x-3)~~\land ~~ W(x)=0~~\Rightarrow ~~(x-\sqrt{3} )\cdot (x+\sqrt{3} )\cdot (2x-3)=0\\\\(x-\sqrt{3} )\cdot (x+\sqrt{3} )\cdot (2x-3)=0\\\\x-\sqrt{3} =0~~\lor~~ x+\sqrt{3}=0~~\lor~~ 2x-3=0\\\\x=\sqrt{3} ~~\lor~~ x=-\sqrt{3}~~\lor~~ x=1,5[/tex]

[tex]zad.3\\\\W(x)=x^{3} -x^{2} -10x-8\\\\\\Wyraz ~~wolny:~~-8\\\\Dzielniki~~wyrazu~~wolnego:~~+1,-1,+2,-2,+4,-4-+8,-8\\\\\\W(-1)=(-1)^{3} -(-1)^{2} -10\cdot (-1)-8=0~~\Rightarrow ~~W(-1)=0\\\\W(-2)=(-2)^{3} -(-2)^{2} -10\cdot (-2)-8=0~~\Rightarrow ~~W(-2)=0\\\\W(-4)=(-4)^{3} -(-4)^{2} -10\cdot (-4)-8=-48~~\Rightarrow ~~W(-4)\neq 0\\\\W(-8)=(-8)^{3} -(-8)^{2} -10\cdot (-8)-8=-504~~\Rightarrow ~~W(-8)\neq 0\\\\W(1)=1^{3} -1^{2} -10\cdot 1-8=-18~~\Rightarrow ~~W(1)\neq 0\\\\[/tex]

[tex]W(2)=2^{3} -2^{2} -10\cdot 2-8=-24~~\Rightarrow ~~W(2)\neq 0\\\\W(4)=4^{3} -4^{2} -10\cdot 4-8=0~~\Rightarrow ~~W(4)=0\\\\W(8)=8^{3} -8^{2} -10\cdot 88-8=360~~\Rightarrow ~~W8)\neq 0\\[/tex]

Odp: Liczby -2, -1 , 4 , które są dzielnikami wyrazu wolnego są pierwiastami wielomianu W(x).

Basetlaviz Basetlaviz · Answer 2

1.

[tex]x^{5}-2x^{3} = 0\\\\x^{3}(x^{2}-2) = 0\\\\x^{3}(x+\sqrt{2})(x-\sqrt{2}) = 0\\\\x^{3} = 0 \ \vee \ x+\sqrt{2} = 0 \ \vee \ x-\sqrt{2} = 0\\\\x = 0 \ \vee \ x = -\sqrt{2} \ \vee \ x = \sqrt{2}\\\\x \in \{-\sqrt{2},0,\sqrt{2}\}[/tex]

2.

[tex]W(x) = 2x^{3}-3x^{2}-6x+9=0 \\\\2x^{3}-3x^{2}-6x+9 = 0\\\\2x^{3}-6x-3x^{2}+9 = 0\\\\2x(x^{2}-3)-3(x^{2}-3) = 0\\\\(x^{2}-3)(2x-3) = 0\\\\(x+\sqrt{3})(x-\sqrt{3})(2x-3) = 0\\\\x+\sqrt{3} = 0 \ \vee \ x-\sqrt{3} = 0 \ \vee \ (2x-3) = 0\\\\x = -\sqrt{3} \ \vee \ x = \sqrt{3} \ \vee \ x = 1,5\\\\x\in\{-\sqrt{3},\sqrt{3},1,5\}[/tex]

3.

[tex]W(x) = x^{3}-x^{2}-10x-8[/tex]

[tex]Wyraz \ wolny: \ -8\\\\Dzielniki \ to: -1, \ 1, \ -2, \ 2, \ -4, \ 4, \ -8, \ 8\\\\W(-1) = (-1)^{3}-(-1)^{2}-10\cdot(-1) - 8 = -1-1+10-8 = \boxed{0}\\\\W(1) = 1^{3}-1^{2}-10\cdot1-8 = 1-1-10-8 = -18\neq 0[/tex]

[tex]W(-2) = (-2)^{3}-(-2)^{2}-10\cdot(-2)-8 = -8-4+20-8 = \boxed{0}\\\\W(2) = 2^{3}-2^{2}-10\cdot2-8 = 8-4-20-8 = -24\neq 0[/tex]

[tex]W(-4) =(-4)^{3}-(-4)^{2}-10\cdot(-4)-8 = -64-16+40-8 = -48\neq 0\\\\W(4) = 4^{3}-4^{2}-10\cdot4-8 = 64-16-40-8 = \boxed{0}[/tex]

[tex]W(-8) = (-8)^{3}-(-8)^{2}-10\cdot(-8)-8 = -512-64+80-8 = -504\neq 0\\\\W(8) = 8^{3}-8^{2}-10\cdot8-8 = 512-64-80-8 = 360\neq 0[/tex]

[tex]Pierwiastkami \ calkowitymi \ sa \ liczby: x = -1, \ x = -2 \ oraz \ x = 4.[/tex]