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Popular Algebra >

expand (x^2-1/x)^{25}

Solution

expand

(x^{2} - \frac{1}{x})^{25}

Solution

x^{50} - 25 x^{47} + 300 x^{44} - 2300 x^{41} + 12650 x^{38} - 53130 x^{35} + 177100 x^{32} - 480700 x^{29} + 1081575 x^{26} - 2042975 x^{23} + 3268760 x^{20} - 4457400 x^{17} + 5200300 x^{14} - 5200300 x^{11} + 4457400 x^{8} - 3268760 x^{5} + 2042975 x^{2} - \frac{1081575}{x} + \frac{480700}{x ^{4}} - \frac{177100}{x ^{7}} + \frac{53130}{x ^{10}} - \frac{12650}{x ^{13}} + \frac{2300}{x ^{16}} - \frac{300}{x ^{19}} + \frac{25}{x ^{22}} - \frac{1}{x ^{25}}

Solution steps

(x^{2} - \frac{1}{x})^{25}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x^{2}, b = - \frac{1}{x}

= i = 0 \sum 25 (i 25) (x^{2})^{(25 - i)} (- \frac{1}{x})^{i}

Expand summation

= \frac{25 !}{0 ! ( 25 - 0 ) !} (x^{2})^{25} (- \frac{1}{x})^{0} + \frac{25 !}{1 ! ( 25 - 1 ) !} (x^{2})^{24} (- \frac{1}{x})^{1} + \frac{25 !}{2 ! ( 25 - 2 ) !} (x^{2})^{23} (- \frac{1}{x})^{2} + \frac{25 !}{3 ! ( 25 - 3 ) !} (x^{2})^{22} (- \frac{1}{x})^{3} + \frac{25 !}{4 ! ( 25 - 4 ) !} (x^{2})^{21} (- \frac{1}{x})^{4} + \frac{25 !}{5 ! ( 25 - 5 ) !} (x^{2})^{20} (- \frac{1}{x})^{5} + \frac{25 !}{6 ! ( 25 - 6 ) !} (x^{2})^{19} (- \frac{1}{x})^{6} + \frac{25 !}{7 ! ( 25 - 7 ) !} (x^{2})^{18} (- \frac{1}{x})^{7} + \frac{25 !}{8 ! ( 25 - 8 ) !} (x^{2})^{17} (- \frac{1}{x})^{8} + \frac{25 !}{9 ! ( 25 - 9 ) !} (x^{2})^{16} (- \frac{1}{x})^{9} + \frac{25 !}{10 ! ( 25 - 10 ) !} (x^{2})^{15} (- \frac{1}{x})^{10} + \frac{25 !}{11 ! ( 25 - 11 ) !} (x^{2})^{14} (- \frac{1}{x})^{11} + \frac{25 !}{12 ! ( 25 - 12 ) !} (x^{2})^{13} (- \frac{1}{x})^{12} + \frac{25 !}{13 ! ( 25 - 13 ) !} (x^{2})^{12} (- \frac{1}{x})^{13} + \frac{25 !}{14 ! ( 25 - 14 ) !} (x^{2})^{11} (- \frac{1}{x})^{14} + \frac{25 !}{15 ! ( 25 - 15 ) !} (x^{2})^{10} (- \frac{1}{x})^{15} + \frac{25 !}{16 ! ( 25 - 16 ) !} (x^{2})^{9} (- \frac{1}{x})^{16} + \frac{25 !}{17 ! ( 25 - 17 ) !} (x^{2})^{8} (- \frac{1}{x})^{17} + \frac{25 !}{18 ! ( 25 - 18 ) !} (x^{2})^{7} (- \frac{1}{x})^{18} + \frac{25 !}{19 ! ( 25 - 19 ) !} (x^{2})^{6} (- \frac{1}{x})^{19} + \frac{25 !}{20 ! ( 25 - 20 ) !} (x^{2})^{5} (- \frac{1}{x})^{20} + \frac{25 !}{21 ! ( 25 - 21 ) !} (x^{2})^{4} (- \frac{1}{x})^{21} + \frac{25 !}{22 ! ( 25 - 22 ) !} (x^{2})^{3} (- \frac{1}{x})^{22} + \frac{25 !}{23 ! ( 25 - 23 ) !} (x^{2})^{2} (- \frac{1}{x})^{23} + \frac{25 !}{24 ! ( 25 - 24 ) !} (x^{2})^{1} (- \frac{1}{x})^{24} + \frac{25 !}{25 ! ( 25 - 25 ) !} (x^{2})^{0} (- \frac{1}{x})^{25}

Simplify

\frac{25 !}{0 ! ( 25 - 0 ) !} (x^{2})^{25} (- \frac{1}{x})^{0} : x^{50}

Simplify

\frac{25 !}{1 ! ( 25 - 1 ) !} (x^{2})^{24} (- \frac{1}{x})^{1} : - 25 x^{47}

Simplify

\frac{25 !}{2 ! ( 25 - 2 ) !} (x^{2})^{23} (- \frac{1}{x})^{2} : 300 x^{44}

Simplify

\frac{25 !}{3 ! ( 25 - 3 ) !} (x^{2})^{22} (- \frac{1}{x})^{3} : - 2300 x^{41}

Simplify

\frac{25 !}{4 ! ( 25 - 4 ) !} (x^{2})^{21} (- \frac{1}{x})^{4} : 12650 x^{38}

Simplify

\frac{25 !}{5 ! ( 25 - 5 ) !} (x^{2})^{20} (- \frac{1}{x})^{5} : - 53130 x^{35}

Simplify

\frac{25 !}{6 ! ( 25 - 6 ) !} (x^{2})^{19} (- \frac{1}{x})^{6} : 177100 x^{32}

Simplify

\frac{25 !}{7 ! ( 25 - 7 ) !} (x^{2})^{18} (- \frac{1}{x})^{7} : - 480700 x^{29}

Simplify

\frac{25 !}{8 ! ( 25 - 8 ) !} (x^{2})^{17} (- \frac{1}{x})^{8} : 1081575 x^{26}

Simplify

\frac{25 !}{9 ! ( 25 - 9 ) !} (x^{2})^{16} (- \frac{1}{x})^{9} : - 2042975 x^{23}

Simplify

\frac{25 !}{10 ! ( 25 - 10 ) !} (x^{2})^{15} (- \frac{1}{x})^{10} : 3268760 x^{20}

Simplify

\frac{25 !}{11 ! ( 25 - 11 ) !} (x^{2})^{14} (- \frac{1}{x})^{11} : - 4457400 x^{17}

Simplify

\frac{25 !}{12 ! ( 25 - 12 ) !} (x^{2})^{13} (- \frac{1}{x})^{12} : 5200300 x^{14}

Simplify

\frac{25 !}{13 ! ( 25 - 13 ) !} (x^{2})^{12} (- \frac{1}{x})^{13} : - 5200300 x^{11}

Simplify

\frac{25 !}{14 ! ( 25 - 14 ) !} (x^{2})^{11} (- \frac{1}{x})^{14} : 4457400 x^{8}

Simplify

\frac{25 !}{15 ! ( 25 - 15 ) !} (x^{2})^{10} (- \frac{1}{x})^{15} : - 3268760 x^{5}

Simplify

\frac{25 !}{16 ! ( 25 - 16 ) !} (x^{2})^{9} (- \frac{1}{x})^{16} : 2042975 x^{2}

Simplify

\frac{25 !}{17 ! ( 25 - 17 ) !} (x^{2})^{8} (- \frac{1}{x})^{17} : - \frac{1081575}{x}

Simplify

\frac{25 !}{18 ! ( 25 - 18 ) !} (x^{2})^{7} (- \frac{1}{x})^{18} : \frac{480700}{x ^{4}}

Simplify

\frac{25 !}{19 ! ( 25 - 19 ) !} (x^{2})^{6} (- \frac{1}{x})^{19} : - \frac{177100}{x ^{7}}

Simplify

\frac{25 !}{20 ! ( 25 - 20 ) !} (x^{2})^{5} (- \frac{1}{x})^{20} : \frac{53130}{x ^{10}}

Simplify

\frac{25 !}{21 ! ( 25 - 21 ) !} (x^{2})^{4} (- \frac{1}{x})^{21} : - \frac{12650}{x ^{13}}

Simplify

\frac{25 !}{22 ! ( 25 - 22 ) !} (x^{2})^{3} (- \frac{1}{x})^{22} : \frac{2300}{x ^{16}}

Simplify

\frac{25 !}{23 ! ( 25 - 23 ) !} (x^{2})^{2} (- \frac{1}{x})^{23} : - \frac{300}{x ^{19}}

Simplify

\frac{25 !}{24 ! ( 25 - 24 ) !} (x^{2})^{1} (- \frac{1}{x})^{24} : \frac{25}{x ^{22}}

Simplify

\frac{25 !}{25 ! ( 25 - 25 ) !} (x^{2})^{0} (- \frac{1}{x})^{25} : - \frac{1}{x ^{25}}

= x^{50} - 25 x^{47} + 300 x^{44} - 2300 x^{41} + 12650 x^{38} - 53130 x^{35} + 177100 x^{32} - 480700 x^{29} + 1081575 x^{26} - 2042975 x^{23} + 3268760 x^{20} - 4457400 x^{17} + 5200300 x^{14} - 5200300 x^{11} + 4457400 x^{8} - 3268760 x^{5} + 2042975 x^{2} - \frac{1081575}{x} + \frac{480700}{x ^{4}} - \frac{177100}{x ^{7}} + \frac{53130}{x ^{10}} - \frac{12650}{x ^{13}} + \frac{2300}{x ^{16}} - \frac{300}{x ^{19}} + \frac{25}{x ^{22}} - \frac{1}{x ^{25}}

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