Objętość prostopadłościanu o krawędziach a, b, c :
V = a × b × c
[tex]a)~~a=x-1,~~b=x+1,~~c=x\\\\V=a\cdot b\cdot c=(x-1)\cdot (x+1)\cdot x =x\cdot (x^{2} -1)=x^{3} -x\\\\a)~~a=x+1,~~b=x+2,~~c=x+3\\\\V=a\cdot b\cdot c=(x+1)\cdot (x+2)\cdot (x+3)=(x^{2} +2x+x+2)\cdot (x+3)=(x^{2} +3x+2)\cdot (x+3)=x^{3} +3x^{2} +3x^{2} +9x+2x+6=x^{3} +6x^{2} +11x+6\\\\c)~~a=2x+1,~~b=\frac{1}{2} x+1,~~c=2x-1\\\\V=a\cdot b\cdot c=(2x+1)\cdot (\frac{1}{2} x+1)\cdot (2x-1)=(x^{2} +2x+\frac{1}{2} x+1)\cdot (2x-1)=(x^{2} +2\frac{1}{2} x+1)\cdot (2x-1)=(x^{2} +\frac{5}{2} x+1)\cdot (2x-1)=[/tex]
[tex]=2x^{3} -x^{2} +5x^{2} -2\frac{1}{2} x+2x-1=2x^{3} +4x^{2} -\frac{1}{2} x-1\\\\d)~~a=x+3,~~b=x+3,~~c=x^{2} -9\\\\V=a\cdot b\cdot c=(x+3)\cdot (x+3)\cdot (x^{2} -9)=(x+3)^{2} \cdot (x^{2} -9)=(x^{2} +6x+9)\cdot (x^{2} -9)=x^{4} -9x^{2} +6x^{3} -54x+9x^{2} -81=x^{4} +6x^{3} -54x-81[/tex]
korzystam ze wzorów:
[tex]x^{n} \cdot x^{m} =x^{n+m} \\\\(x-y)\cdot (x+y)= x^{2} -y^{2} \\\\(x+y)\cdot (x+y)=(x+y)^{2} =x^{2} +2xy+y^{2}[/tex]
