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Phổ biến Đại số >

khai triển (3x-(x^2)/3)^{13}

Lời Giải

khai triển

(3 x - \frac{x ^{2}}{3})^{13}

Lời Giải

1594323 x^{13} - 2302911 x^{14} + 1535274 x^{15} - 625482 x^{16} + 173745 x^{17} - 34749 x^{18} + 5148 x^{19} - 572 x^{20} + \frac{143 x ^{21}}{3} - \frac{715 x ^{22}}{243} + \frac{286 x ^{23}}{2187} - \frac{26 x ^{24}}{6561} + \frac{13 x ^{25}}{177147} - \frac{x ^{26}}{1594323}

Các bước giải pháp

(3 x - \frac{x ^{2}}{3})^{13}

Áp dụng định lý nhị thức:

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

a = 3 x, b = - \frac{x ^{2}}{3}

= i = 0 \sum 13 (i 13) (3 x)^{(13 - i)} (- \frac{x ^{2}}{3})^{i}

Khai triển phép tổng

= \frac{13 !}{0 ! ( 13 - 0 ) !} (3 x)^{13} (- \frac{x ^{2}}{3})^{0} + \frac{13 !}{1 ! ( 13 - 1 ) !} (3 x)^{12} (- \frac{x ^{2}}{3})^{1} + \frac{13 !}{2 ! ( 13 - 2 ) !} (3 x)^{11} (- \frac{x ^{2}}{3})^{2} + \frac{13 !}{3 ! ( 13 - 3 ) !} (3 x)^{10} (- \frac{x ^{2}}{3})^{3} + \frac{13 !}{4 ! ( 13 - 4 ) !} (3 x)^{9} (- \frac{x ^{2}}{3})^{4} + \frac{13 !}{5 ! ( 13 - 5 ) !} (3 x)^{8} (- \frac{x ^{2}}{3})^{5} + \frac{13 !}{6 ! ( 13 - 6 ) !} (3 x)^{7} (- \frac{x ^{2}}{3})^{6} + \frac{13 !}{7 ! ( 13 - 7 ) !} (3 x)^{6} (- \frac{x ^{2}}{3})^{7} + \frac{13 !}{8 ! ( 13 - 8 ) !} (3 x)^{5} (- \frac{x ^{2}}{3})^{8} + \frac{13 !}{9 ! ( 13 - 9 ) !} (3 x)^{4} (- \frac{x ^{2}}{3})^{9} + \frac{13 !}{10 ! ( 13 - 10 ) !} (3 x)^{3} (- \frac{x ^{2}}{3})^{10} + \frac{13 !}{11 ! ( 13 - 11 ) !} (3 x)^{2} (- \frac{x ^{2}}{3})^{11} + \frac{13 !}{12 ! ( 13 - 12 ) !} (3 x)^{1} (- \frac{x ^{2}}{3})^{12} + \frac{13 !}{13 ! ( 13 - 13 ) !} (3 x)^{0} (- \frac{x ^{2}}{3})^{13}

Rút gọn

\frac{13 !}{0 ! ( 13 - 0 ) !} (3 x)^{13} (- \frac{x ^{2}}{3})^{0} : 1594323 x^{13}

Rút gọn

\frac{13 !}{1 ! ( 13 - 1 ) !} (3 x)^{12} (- \frac{x ^{2}}{3})^{1} : - 2302911 x^{14}

Rút gọn

\frac{13 !}{2 ! ( 13 - 2 ) !} (3 x)^{11} (- \frac{x ^{2}}{3})^{2} : 1535274 x^{15}

Rút gọn

\frac{13 !}{3 ! ( 13 - 3 ) !} (3 x)^{10} (- \frac{x ^{2}}{3})^{3} : - 625482 x^{16}

Rút gọn

\frac{13 !}{4 ! ( 13 - 4 ) !} (3 x)^{9} (- \frac{x ^{2}}{3})^{4} : 173745 x^{17}

Rút gọn

\frac{13 !}{5 ! ( 13 - 5 ) !} (3 x)^{8} (- \frac{x ^{2}}{3})^{5} : - 34749 x^{18}

Rút gọn

\frac{13 !}{6 ! ( 13 - 6 ) !} (3 x)^{7} (- \frac{x ^{2}}{3})^{6} : 5148 x^{19}

Rút gọn

\frac{13 !}{7 ! ( 13 - 7 ) !} (3 x)^{6} (- \frac{x ^{2}}{3})^{7} : - 572 x^{20}

Rút gọn

\frac{13 !}{8 ! ( 13 - 8 ) !} (3 x)^{5} (- \frac{x ^{2}}{3})^{8} : \frac{143 x ^{21}}{3}

Rút gọn

\frac{13 !}{9 ! ( 13 - 9 ) !} (3 x)^{4} (- \frac{x ^{2}}{3})^{9} : - \frac{715 x ^{22}}{243}

Rút gọn

\frac{13 !}{10 ! ( 13 - 10 ) !} (3 x)^{3} (- \frac{x ^{2}}{3})^{10} : \frac{286 x ^{23}}{2187}

Rút gọn

\frac{13 !}{11 ! ( 13 - 11 ) !} (3 x)^{2} (- \frac{x ^{2}}{3})^{11} : - \frac{26 x ^{24}}{6561}

Rút gọn

\frac{13 !}{12 ! ( 13 - 12 ) !} (3 x)^{1} (- \frac{x ^{2}}{3})^{12} : \frac{13 x ^{25}}{177147}

Rút gọn

\frac{13 !}{13 ! ( 13 - 13 ) !} (3 x)^{0} (- \frac{x ^{2}}{3})^{13} : - \frac{x ^{26}}{1594323}

= 1594323 x^{13} - 2302911 x^{14} + 1535274 x^{15} - 625482 x^{16} + 173745 x^{17} - 34749 x^{18} + 5148 x^{19} - 572 x^{20} + \frac{143 x ^{21}}{3} - \frac{715 x ^{22}}{243} + \frac{286 x ^{23}}{2187} - \frac{26 x ^{24}}{6561} + \frac{13 x ^{25}}{177147} - \frac{x ^{26}}{1594323}

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