[tex]a) \\ \\ ( {a}^{3} - 1)( {a}^{6} + {a}^{3} + 1) = \\ \\ {a}^{3}( {a}^{6} + {a}^{3} + 1) - ( {a}^{6} + {a}^{3} + 1) = \\ \\ {a}^{9} + {a}^{6} + {a}^{3} - {a}^{6} - {a}^{3} - 1 = \\ \\ {a}^{9} + ( {a}^{6} - {a}^{6}) + ( {a}^{3} - {a}^{3}) - 1 = \\ \\ {a}^{9} - 1 [/tex]
b)
(m-1)(m²+m+1)=
m(m²+m+1)-(m²+m+1)=
m³+m²+m-m²-m-1=
m³+(m²-m²)+(m-m)-1=
m³-1
[tex]c) \\ \\ ( {a}^{2} + 1)( {a}^{4} - {a}^{2} + 1) = \\ \\ {a}^{2}( {a}^{4} - {a}^{2} + 1) + {a}^{4} - {a}^{2} + 1 = \\ \\ {a}^{6} - {a}^{4} + {a}^{2} + {a}^{4} - {a}^{2} + 1 = \\ \\ {a}^{6} + ( - {a}^{4} + {a}^{4}) + ( {a}^{2} - {a}^{2}) + 1 = \\ \\ {a}^{6} + 1 [/tex]
d)
(2a+3b)(4a²-6ab+9b²)=
2a(4a²-6ab+9b²)+3b(4a²-6ab+9b²)=
8a³-12a²b+18ab²+3b(4a²-6ab+9b²)=
8a³-12a²b+18ab²+12ba²-18b²a+27b³=
8a³+(-12a²b+12a²b)+(18ab²-18ab²)+27b³=
8a³+27b³
