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受欢迎的代数 >

展开 (2x+3)^{12}

解答

展开

(2 x + 3)^{12}

解答

4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441

求解步骤

(2 x + 3)^{12}

使用二项式定理:

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

a = 2 x, b = 3

= i = 0 \sum 12 (i 12) (2 x)^{(12 - i)} \cdot 3^{i}

展开求和

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

化简

\frac{12 !}{0 ! ( 12 - 0 ) !} (2 x)^{12} \cdot 3^{0} : 4096 x^{12}

化简

\frac{12 !}{1 ! ( 12 - 1 ) !} (2 x)^{11} \cdot 3^{1} : 73728 x^{11}

化简

\frac{12 !}{2 ! ( 12 - 2 ) !} (2 x)^{10} \cdot 3^{2} : 608256 x^{10}

化简

\frac{12 !}{3 ! ( 12 - 3 ) !} (2 x)^{9} \cdot 3^{3} : 3041280 x^{9}

化简

\frac{12 !}{4 ! ( 12 - 4 ) !} (2 x)^{8} \cdot 3^{4} : 10264320 x^{8}

化简

\frac{12 !}{5 ! ( 12 - 5 ) !} (2 x)^{7} \cdot 3^{5} : 24634368 x^{7}

化简

\frac{12 !}{6 ! ( 12 - 6 ) !} (2 x)^{6} \cdot 3^{6} : 43110144 x^{6}

化简

\frac{12 !}{7 ! ( 12 - 7 ) !} (2 x)^{5} \cdot 3^{7} : 55427328 x^{5}

化简

\frac{12 !}{8 ! ( 12 - 8 ) !} (2 x)^{4} \cdot 3^{8} : 51963120 x^{4}

化简

\frac{12 !}{9 ! ( 12 - 9 ) !} (2 x)^{3} \cdot 3^{9} : 34642080 x^{3}

化简

\frac{12 !}{10 ! ( 12 - 10 ) !} (2 x)^{2} \cdot 3^{10} : 15588936 x^{2}

化简

\frac{12 !}{11 ! ( 12 - 11 ) !} (2 x)^{1} \cdot 3^{11} : 4251528 x

化简

\frac{12 !}{12 ! ( 12 - 12 ) !} (2 x)^{0} \cdot 3^{12} : 531441

= 4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441

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