expand ((x^5+3)^{10})/(10)

פתרון

expand

\frac{( x ^{5} + 3 ) ^{10}}{10}

\frac{x ^{50}}{10} + 3 x^{45} + \frac{81 x ^{40}}{2} + 324 x^{35} + 1701 x^{30} + \frac{30618 x ^{25}}{5} + 15309 x^{20} + 26244 x^{15} + \frac{59049 x ^{10}}{2} + 19683 x^{5} + \frac{59049}{10}

צעדי פתרון

\frac{( x ^{5} + 3 ) ^{10}}{10}

(x^{5} + 3)^{10} = x^{50} + 30 x^{45} + 405 x^{40} + 3240 x^{35} + 17010 x^{30} + 61236 x^{25} + 153090 x^{20} + 262440 x^{15} + 295245 x^{10} + 196830 x^{5} + 59049

= \frac{x ^{50} + 30 x ^{45} + 405 x ^{40} + 3240 x ^{35} + 17010 x ^{30} + 61236 x ^{25} + 153090 x ^{20} + 262440 x ^{15} + 295245 x ^{10} + 196830 x ^{5} + 59049}{10}

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

: השתמש בתכונת השברים הבאה

\frac{x ^{50} + 30 x ^{45} + 405 x ^{40} + 3240 x ^{35} + 17010 x ^{30} + 61236 x ^{25} + 153090 x ^{20} + 262440 x ^{15} + 295245 x ^{10} + 196830 x ^{5} + 59049}{10} = \frac{x ^{50}}{10} + \frac{30 x ^{45}}{10} + \frac{405 x ^{40}}{10} + \frac{3240 x ^{35}}{10} + \frac{17010 x ^{30}}{10} + \frac{61236 x ^{25}}{10} + \frac{153090 x ^{20}}{10} + \frac{262440 x ^{15}}{10} + \frac{295245 x ^{10}}{10} + \frac{196830 x ^{5}}{10} + \frac{59049}{10}

= \frac{x ^{50}}{10} + \frac{30 x ^{45}}{10} + \frac{405 x ^{40}}{10} + \frac{3240 x ^{35}}{10} + \frac{17010 x ^{30}}{10} + \frac{61236 x ^{25}}{10} + \frac{153090 x ^{20}}{10} + \frac{262440 x ^{15}}{10} + \frac{295245 x ^{10}}{10} + \frac{196830 x ^{5}}{10} + \frac{59049}{10}

\frac{30 x ^{45}}{10} = 3 x^{45}

\frac{405 x ^{40}}{10} = \frac{81 x ^{40}}{2}

\frac{3240 x ^{35}}{10} = 324 x^{35}

\frac{17010 x ^{30}}{10} = 1701 x^{30}

\frac{61236 x ^{25}}{10} = \frac{30618 x ^{25}}{5}

\frac{153090 x ^{20}}{10} = 15309 x^{20}

\frac{262440 x ^{15}}{10} = 26244 x^{15}

\frac{295245 x ^{10}}{10} = \frac{59049 x ^{10}}{2}

\frac{196830 x ^{5}}{10} = 19683 x^{5}

= \frac{x ^{50}}{10} + 3 x^{45} + \frac{81 x ^{40}}{2} + 324 x^{35} + 1701 x^{30} + \frac{30618 x ^{25}}{5} + 15309 x^{20} + 26244 x^{15} + \frac{59049 x ^{10}}{2} + 19683 x^{5} + \frac{59049}{10}

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