simplificar (x^{100}-1)/(x+1)

Solução

simplificar

\frac{x ^{100} - 1}{x + 1}

(x^{2} + 1) (x^{48} - x^{46} + x^{44} - x^{42} + x^{40} - x^{38} + x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1) (x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1) (x - 1) (x^{24} + x^{23} + x^{22} + x^{21} + x^{20} + x^{19} + x^{18} + x^{17} + x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)

Passos da solução

\frac{x ^{100} - 1}{x + 1}

Aplicar as propriedades dos expoentes

= \frac{( x ^{50} ) ^{2} - 1 ^{2}}{x + 1}

Aplicar a regra da diferença de quadrados:

x^{2} - y^{2} = (x + y) (x - y)

(x^{50})^{2} - 1^{2} = (x^{50} + 1) (x^{50} - 1)

= \frac{( x ^{50} + 1 ) ( x ^{50} - 1 )}{x + 1}

Fatorar

x^{50} + 1 : (x^{2} + 1) (x^{48} - x^{46} + x^{44} - x^{42} + x^{40} - x^{38} + x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1)

= \frac{( x ^{2} + 1 ) ( x ^{48} - x ^{46} + x ^{44} - x ^{42} + x ^{40} - x ^{38} + x ^{36} - x ^{34} + x ^{32} - x ^{30} + x ^{28} - x ^{26} + x ^{24} - x ^{22} + x ^{20} - x ^{18} + x ^{16} - x ^{14} + x ^{12} - x ^{10} + x ^{8} - x ^{6} + x ^{4} - x ^{2} + 1 ) ( x ^{50} - 1 )}{x + 1}

Aplicar as propriedades dos expoentes

= \frac{( x ^{2} + 1 ) ( x ^{48} - x ^{46} + x ^{44} - x ^{42} + x ^{40} - x ^{38} + x ^{36} - x ^{34} + x ^{32} - x ^{30} + x ^{28} - x ^{26} + x ^{24} - x ^{22} + x ^{20} - x ^{18} + x ^{16} - x ^{14} + x ^{12} - x ^{10} + x ^{8} - x ^{6} + x ^{4} - x ^{2} + 1 ) ( ( x ^{25} ) ^{2} - 1 ^{2} )}{x + 1}

Aplicar a regra da diferença de quadrados:

x^{2} - y^{2} = (x + y) (x - y)

(x^{25})^{2} - 1^{2} = (x^{25} + 1) (x^{25} - 1)

= \frac{( x ^{2} + 1 ) ( x ^{48} - x ^{46} + x ^{44} - x ^{42} + x ^{40} - x ^{38} + x ^{36} - x ^{34} + x ^{32} - x ^{30} + x ^{28} - x ^{26} + x ^{24} - x ^{22} + x ^{20} - x ^{18} + x ^{16} - x ^{14} + x ^{12} - x ^{10} + x ^{8} - x ^{6} + x ^{4} - x ^{2} + 1 ) ( x ^{25} + 1 ) ( x ^{25} - 1 )}{x + 1}

Fatorar

x^{25} + 1 : (x + 1) (x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1)

= \frac{( x ^{2} + 1 ) ( x ^{48} - x ^{46} + x ^{44} - x ^{42} + x ^{40} - x ^{38} + x ^{36} - x ^{34} + x ^{32} - x ^{30} + x ^{28} - x ^{26} + x ^{24} - x ^{22} + x ^{20} - x ^{18} + x ^{16} - x ^{14} + x ^{12} - x ^{10} + x ^{8} - x ^{6} + x ^{4} - x ^{2} + 1 ) ( x + 1 ) ( x ^{24} - x ^{23} + x ^{22} - x ^{21} + x ^{20} - x ^{19} + x ^{18} - x ^{17} + x ^{16} - x ^{15} + x ^{14} - x ^{13} + x ^{12} - x ^{11} + x ^{10} - x ^{9} + x ^{8} - x ^{7} + x ^{6} - x ^{5} + x ^{4} - x ^{3} + x ^{2} - x + 1 ) ( x ^{25} - 1 )}{x + 1}

Fatorar

x^{25} - 1 : (x - 1) (x^{24} + x^{23} + x^{22} + x^{21} + x^{20} + x^{19} + x^{18} + x^{17} + x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)

= \frac{( x ^{2} + 1 ) ( x ^{48} - x ^{46} + x ^{44} - x ^{42} + x ^{40} - x ^{38} + x ^{36} - x ^{34} + x ^{32} - x ^{30} + x ^{28} - x ^{26} + x ^{24} - x ^{22} + x ^{20} - x ^{18} + x ^{16} - x ^{14} + x ^{12} - x ^{10} + x ^{8} - x ^{6} + x ^{4} - x ^{2} + 1 ) ( x + 1 ) ( x ^{24} - x ^{23} + x ^{22} - x ^{21} + x ^{20} - x ^{19} + x ^{18} - x ^{17} + x ^{16} - x ^{15} + x ^{14} - x ^{13} + x ^{12} - x ^{11} + x ^{10} - x ^{9} + x ^{8} - x ^{7} + x ^{6} - x ^{5} + x ^{4} - x ^{3} + x ^{2} - x + 1 ) ( x - 1 ) ( x ^{24} + x ^{23} + x ^{22} + x ^{21} + x ^{20} + x ^{19} + x ^{18} + x ^{17} + x ^{16} + x ^{15} + x ^{14} + x ^{13} + x ^{12} + x ^{11} + x ^{10} + x ^{9} + x ^{8} + x ^{7} + x ^{6} + x ^{5} + x ^{4} + x ^{3} + x ^{2} + x + 1 )}{x + 1}

Eliminar o fator comum:

x + 1

= (x^{2} + 1) (x^{48} - x^{46} + x^{44} - x^{42} + x^{40} - x^{38} + x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1) (x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1) (x - 1) (x^{24} + x^{23} + x^{22} + x^{21} + x^{20} + x^{19} + x^{18} + x^{17} + x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)

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