展開する (1+2/3 x)^9(3+9x)^2

解

展開する

(1 + \frac{2}{3} x)^{9} (3 + 9 x)^{2}

9 + 108 x + 549 x^{2} + 1574 x^{3} + 2864 x^{4} + \frac{10528 x ^{5}}{3} + \frac{80416 x ^{6}}{27} + \frac{47552 x ^{7}}{27} + \frac{57856 x ^{8}}{81} + \frac{415232 x ^{9}}{2187} + \frac{21760 x ^{10}}{729} + \frac{512 x ^{11}}{243}

解答ステップ

(1 + \frac{2}{3} x)^{9} (3 + 9 x)^{2}

拡張

(3 + 9 x)^{2} : 9 + 54 x + 81 x^{2}

= (1 + \frac{2}{3} x)^{9} (9 + 54 x + 81 x^{2})

分配法則を適用する:

m (a + b + c) = ma + mb + m c

(1 + \frac{2}{3} x)^{9} (9 + 54 x + 81 x^{2}) = (1 + \frac{2}{3} x)^{9} \cdot 9 + (1 + \frac{2}{3} x)^{9} \cdot 54 x + (1 + \frac{2}{3} x)^{9} \cdot 81 x^{2}

= (1 + \frac{2}{3} x)^{9} \cdot 9 + (1 + \frac{2}{3} x)^{9} \cdot 54 x + (1 + \frac{2}{3} x)^{9} \cdot 81 x^{2}

= 9 (1 + \frac{2}{3} x)^{9} + 54 x (1 + \frac{2}{3} x)^{9} + 81 x^{2} (1 + \frac{2}{3} x)^{9}

拡張

9 (1 + \frac{2}{3} x)^{9} : 9 + 54 x + 144 x^{2} + 224 x^{3} + 224 x^{4} + \frac{448 x ^{5}}{3} + \frac{1792 x ^{6}}{27} + \frac{512 x ^{7}}{27} + \frac{256 x ^{8}}{81} + \frac{512 x ^{9}}{2187}

= 9 + 54 x + 144 x^{2} + 224 x^{3} + 224 x^{4} + \frac{448 x ^{5}}{3} + \frac{1792 x ^{6}}{27} + \frac{512 x ^{7}}{27} + \frac{256 x ^{8}}{81} + \frac{512 x ^{9}}{2187} + 54 x (1 + \frac{2}{3} x)^{9} + 81 x^{2} (1 + \frac{2}{3} x)^{9}

拡張

54 x (1 + \frac{2}{3} x)^{9} : 54 x + 324 x^{2} + 864 x^{3} + 1344 x^{4} + 1344 x^{5} + 896 x^{6} + \frac{3584 x ^{7}}{9} + \frac{1024 x ^{8}}{9} + \frac{512 x ^{9}}{27} + \frac{1024 x ^{10}}{729}

= 9 + 54 x + 144 x^{2} + 224 x^{3} + 224 x^{4} + \frac{448 x ^{5}}{3} + \frac{1792 x ^{6}}{27} + \frac{512 x ^{7}}{27} + \frac{256 x ^{8}}{81} + \frac{512 x ^{9}}{2187} + 54 x + 324 x^{2} + 864 x^{3} + 1344 x^{4} + 1344 x^{5} + 896 x^{6} + \frac{3584 x ^{7}}{9} + \frac{1024 x ^{8}}{9} + \frac{512 x ^{9}}{27} + \frac{1024 x ^{10}}{729} + 81 x^{2} (1 + \frac{2}{3} x)^{9}

拡張

81 x^{2} (1 + \frac{2}{3} x)^{9} : 81 x^{2} + 486 x^{3} + 1296 x^{4} + 2016 x^{5} + 2016 x^{6} + 1344 x^{7} + \frac{1792 x ^{8}}{3} + \frac{512 x ^{9}}{3} + \frac{256 x ^{10}}{9} + \frac{512 x ^{11}}{243}

= 9 + 54 x + 144 x^{2} + 224 x^{3} + 224 x^{4} + \frac{448 x ^{5}}{3} + \frac{1792 x ^{6}}{27} + \frac{512 x ^{7}}{27} + \frac{256 x ^{8}}{81} + \frac{512 x ^{9}}{2187} + 54 x + 324 x^{2} + 864 x^{3} + 1344 x^{4} + 1344 x^{5} + 896 x^{6} + \frac{3584 x ^{7}}{9} + \frac{1024 x ^{8}}{9} + \frac{512 x ^{9}}{27} + \frac{1024 x ^{10}}{729} + 81 x^{2} + 486 x^{3} + 1296 x^{4} + 2016 x^{5} + 2016 x^{6} + 1344 x^{7} + \frac{1792 x ^{8}}{3} + \frac{512 x ^{9}}{3} + \frac{256 x ^{10}}{9} + \frac{512 x ^{11}}{243}

簡素化

9 + 54 x + 144 x^{2} + 224 x^{3} + 224 x^{4} + \frac{448 x ^{5}}{3} + \frac{1792 x ^{6}}{27} + \frac{512 x ^{7}}{27} + \frac{256 x ^{8}}{81} + \frac{512 x ^{9}}{2187} + 54 x + 324 x^{2} + 864 x^{3} + 1344 x^{4} + 1344 x^{5} + 896 x^{6} + \frac{3584 x ^{7}}{9} + \frac{1024 x ^{8}}{9} + \frac{512 x ^{9}}{27} + \frac{1024 x ^{10}}{729} + 81 x^{2} + 486 x^{3} + 1296 x^{4} + 2016 x^{5} + 2016 x^{6} + 1344 x^{7} + \frac{1792 x ^{8}}{3} + \frac{512 x ^{9}}{3} + \frac{256 x ^{10}}{9} + \frac{512 x ^{11}}{243} : 9 + 108 x + 549 x^{2} + 1574 x^{3} + 2864 x^{4} + \frac{10528 x ^{5}}{3} + \frac{80416 x ^{6}}{27} + \frac{47552 x ^{7}}{27} + \frac{57856 x ^{8}}{81} + \frac{415232 x ^{9}}{2187} + \frac{21760 x ^{10}}{729} + \frac{512 x ^{11}}{243}

= 9 + 108 x + 549 x^{2} + 1574 x^{3} + 2864 x^{4} + \frac{10528 x ^{5}}{3} + \frac{80416 x ^{6}}{27} + \frac{47552 x ^{7}}{27} + \frac{57856 x ^{8}}{81} + \frac{415232 x ^{9}}{2187} + \frac{21760 x ^{10}}{729} + \frac{512 x ^{11}}{243}