3.
W(-1)=0
[tex]\begin{array}{lll}(3x^3+x^2-5x-3)&:& (x+1)=3x^2-2x-3\\\underline{-(-3x^3+3x^2)} & & \\\qquad -2x^2-5x-3 & & \\\qquad \ \ \underline{-(-2x^2-2x)} & &\\\qquad \qquad \qquad -3x-3 & & \\\qquad \qquad \quad \underline{-(-3x-3)} & & \\\qquad \qquad \qquad \qquad R=0 & & \\\end{array}\\[/tex]
[tex]W(x)=(x+1)(3x^2-2x-3)\\\\(x+1)(3x^2-2x-3)=0\\\\x+1=0\ \vee \ 3x^2-2x-3=0\\\\\Delta=40\rightarrow \sqrt{\Delta}=2\sqrt{10}\\\\x+1=0\ \vee \ 3x^2-2x-3=0\\\\x_1=-1\ \vee \ x_2=\frac{2-2\sqrt{10}}{6}\ \vee \ x_{3}=\frac{2+2\sqrt{10}}{6}\\\\\frac{2-2\sqrt{10}}{6}\approx-0,72\ \ \ , \ \ \frac{2+2\sqrt{10}}{6}\approx1,39\\\\[/tex]
X3 to największy pierwiastek wielomianu
4
[tex]x\ \ , \ \ x+3\ \ , \ \ x-4\\\\x(x+3)(x-4)=-180\\\\x(x^2-x-12)=-180\\\\x^3-x^2-12x+180=0\\\\W(x)=x^3-x^2-12x+180\\\\W(-6)=0\\\\\\\begin{array}{lll}(x^3-x^2-12x+180)&:& (x+6)=x^2-7x+30\\\underline{-(x^3+6x^2)} & & \\\qquad -7x^2-12x+180& & \\\qquad \ \ \underline{-(-7x^2-42x)} & &\\\qquad \qquad \qquad 30x+180& & \\\qquad \qquad \quad \underline{-(30x+180)} & & \\\qquad \qquad \qquad \qquad R=0 & & \\\end{array}[/tex]
D: x ∈ C
[tex](x+6)(x^2-7x+30)=0\\\\x+6=0\ \vee \ x^2-7x+30=0\\\\\Delta=-71\\\\x+6=0\ \vee \ x^2-7x+30=0\\\\x=-6\ \vee \ x\in \phi\\\\[/tex]
Odp. Te liczby to: -6, -3, -10.
5.
[tex]4x^3+7x^2-5x-6=0\\\\W(1)=0\\\\\\\begin{array}{lll}(4x^3+7x^2-5x-6)&:&(x-1)=4x^2+11x+6\\\underline{-(4x^3-4x^2)}&&\\\qquad11x^2-5x-6&&\\\qquad \ \ \underline{-(11x^2-11x)}&&\\\qquad \qquad \qquad 6x-6&&\\\qquad \qquad \quad \underline{-(6x-6)}&&\\\qquad \qquad \qquad \qquad R=0&&\\\end{array}\\[/tex]
[tex](x-1)(4x^2+11x+6)=0\\\\x-1=0\ \vee \ 4x^2+11x+6=0\\\\\Delta=25\rightarrow \sqrt{\Delta}=5\\\\x_1=1\ \vee \ x_2=\frac{-11-5}{8}\ \vee \ x_3=\frac{-11+5}{8}\\\\x_1=1\ \vee \ x_2=-2\ \vee \ x_3=-\frac{3}{4}\\[/tex]