Korzystam ze wzorów skróconego mnożenia:
[tex](a+b)^{3} =a^{3} +3a^{2} b+3ab^{2} +b^{3} \\\\(a-b)^{3} =a^{3} -3a^{2} b+3ab^{2} -b^{3} \\\\a^{3} -b^{3} =(a-b)(a^{2} +ab+b^{2} )\\\\a^{3} +b^{3} =(a-b)(a^{2} -ab+b^{2} )\\\\(a-b)(a+b)=a^{2} -b^{2} \\\\(a-b)^{2} =a^{2} -2ab+b^{2} \\\\(a+b)^{2} =a^{2} +2ab+b^{2}[/tex]
[tex]zad.1\\\\\(x^{5} +x^{2} )^{3} =(x^{5} )^{3} +3(x^{5} )^{2} x^{2} +3x^{5} (x^{2} )^{2} +(x^{2} )^{3} =x^{5\cdot3} +3x^{5\cdot2}x^{2} +3x^{5} x^{2\cdot 2} +x^{2\cdot 3} =x^{15} +3x^{10}x^{2} +3x^{5} x^{4} +x^{6} =x^{15} +3x^{10+2} +3x^{5+4} +x^{6} =x^{15} +3x^{12} +3x^{9} +x^{6} \\\\zad.2\\\\(3x-7)^{3} =(3x)^{3} -3(3x)^{2} \cdot7+3\cdot3x \cdot 7^{2} -7^{3} =27x^{3} -3\cdot 9x^{2} \cdot 7+9x\cdot 49-343=27x^{3} -189x^{2} +441x-343\\\\[/tex]
[tex]zad.3\\\\123 -x^{3} =5^{3} -x^{3} =(5-x)\cdot (5^{2} +5a+x^{2} )= (5-x)\cdot (25 +5a+x^{2} )\\\\zad.4\\\\216+ 8x^{3} = 6^{3} +(2x)^{3} =(6+2x)\cdot (6^{2} -6\cdot 2x +(2x)^{2} )=(6+2x)\cdot (36 -12x +4x^{2} )\\\\[/tex]
[tex]zad.5\\I ~sposob\\(x+2)^{3} -(x-2)^{3} =x^{3} +3x^{2} \cdot 2+3x \cdot 2^{2} +2^{3} -(x^{3} -3x^{2} \cdot 2+3x \cdot 2^{2} -2^{3})=x^{3}+6x^{2} +3x\cdot 4+8-(x^{3}-6x^{2} +3x\cdot 4-8)= x^{3}+6x^{2} +12x+8-(x^{3}-6x^{2} +12x-8)= x^{3}+6x^{2} +12x+8-x^{3}+6x^{2} -12x+8=12x^{2} +16[/tex]
[tex]II~sposob\\(x+2)^{3} -(x-2)^{3}=[x+2-(x-2)]\cdot [(x+2)^{2} +(x+2)(x-2)+(x-2)^{2} ]=(x+2-x+2)\cdot [x^{2} +4x+4+(x^{2} -2^{2})+x^{2} -4x+4]=4\cdot (2x^{2} + 8+x^{2} -4)=4\cdot ( 3x^{2} +4)= 12x^{2} +16[/tex]
