1.
[tex]w(x)=2x^3-x^2+3\\w(1-2\sqrt3)=2*(1-2\sqrt3)^3-(1-2\sqrt3)^2+3\\w(1-2\sqrt3)=2*(1^3-3*1^2*2\sqrt3+3*1*(2\sqrt3)^2-(2\sqrt3)^3)-(1^2-2*1*2\sqrt3)+(2\sqrt3)^2)+3\\w(1-2\sqrt3)=2*(1-6\sqrt3+3*12-24\sqrt3)-(1-4\sqrt3+12)+3\\w(1-2\sqrt3)=2*(37-30\sqrt3)-(13-4\sqrt3)+3\\w(1-2\sqrt3)=74-60\sqrt3-13+4\sqrt3+3\\w(1-2\sqrt3)=61-56\sqrt3+3\\w(1-2\sqrt3)=64-56\sqrt3[/tex]
2.
[tex]w(x)=(3x^2-7x-20)(-x^2+3x-5) \\w(x)=(3x^2-7x-20)-(x^2-3x+5)\\\Delta_1=(-7)^2-4*2*(-20)\\\Delta_1=49+240\\\Delta_1=289\\\sqrt{\Delta_1}=17\\x_1=\frac{7-17}{6}=\frac{-10}6=-\frac53\\x_1=\frac{7+17}6=\frac{24}6=4\\3x^2-7x-20=3(x+\frac53)(x-4)=(3x+5)(x-4)\\\Delta_2=(-3)^2-4*1*5\\\Delta_2=9-20=-11\\w(x)=(3x+5)(x-4)(-x^2+3x-5)=-(3x+5)(x-4)(x^2-3x+5)[/tex]
Wzory skróconego mnożenia:
[tex](a+b)^2=a^2+2ab+c^2\\(a-b)^2=a^2-2ab+b^2\\(a+b)^3=(a+b)(a+b)^2=(a+b)(a^2+2ab+b^2)=a^3+2a^2b+ab^2+a^2b+2ab^2+b^3=a^3+3a^2b+3ab^2+b^3\\(a-b)^3=(a-b)(a-b)^2=(a-b)(a^2-2ab+b^2)=a^3-2a^2b+ab^2-a^2b+2ab^2-b^3=a^3-3a^2b+3ab^2-b^3[/tex]
