展開する (3x^2+2y)^4

解

展開する

(3 x^{2} + 2 y)^{4}

81 x^{8} + 216 x^{6} y + 216 x^{4} y^{2} + 96 x^{2} y^{3} + 16 y^{4}

解答ステップ

(3 x^{2} + 2 y)^{4}

2項定理を適用する:

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

a = 3 x^{2}, b = 2 y

= i = 0 \sum 4 (i 4) (3 x^{2})^{(4 - i)} (2 y)^{i}

総和を展開する

= \frac{4 !}{0 ! ( 4 - 0 ) !} (3 x^{2})^{4} (2 y)^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (3 x^{2})^{3} (2 y)^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (3 x^{2})^{2} (2 y)^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (3 x^{2})^{1} (2 y)^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (3 x^{2})^{0} (2 y)^{4}

簡素化

\frac{4 !}{0 ! ( 4 - 0 ) !} (3 x^{2})^{4} (2 y)^{0} : 81 x^{8}

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} (3 x^{2})^{3} (2 y)^{1} : 216 x^{6} y

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} (3 x^{2})^{2} (2 y)^{2} : 216 x^{4} y^{2}

簡素化

\frac{4 !}{3 ! ( 4 - 3 ) !} (3 x^{2})^{1} (2 y)^{3} : 96 x^{2} y^{3}

簡素化

\frac{4 !}{4 ! ( 4 - 4 ) !} (3 x^{2})^{0} (2 y)^{4} : 16 y^{4}

= 81 x^{8} + 216 x^{6} y + 216 x^{4} y^{2} + 96 x^{2} y^{3} + 16 y^{4}