展開する (1+z+(z^2)/2)^4

解

展開する

(1 + z + \frac{z ^{2}}{2})^{4}

\frac{17 z ^{4}}{2} + 5 z^{5} + 2 z^{6} + 10 z^{3} + 8 z^{2} + 4 z + 1 + \frac{z ^{7}}{2} + \frac{z ^{8}}{16}

解答ステップ

(1 + z + \frac{z ^{2}}{2})^{4}

簡素化

(1 + z + \frac{z ^{2}}{2})^{4} : (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2})

= (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2})

拡張

(1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2}) : 1 + 2 z + 2 z^{2} + z^{3} + \frac{z ^{4}}{4}

= (1 + 2 z + 2 z^{2} + z^{3} + \frac{z ^{4}}{4}) (1 + z + \frac{z ^{2}}{2}) (1 + z + \frac{z ^{2}}{2})

拡張

(1 + 2 z + 2 z^{2} + z^{3} + \frac{z ^{4}}{4}) (1 + z + \frac{z ^{2}}{2}) : \frac{9 z ^{4}}{4} + 4 z^{3} + \frac{9 z ^{2}}{2} + 3 z + 1 + \frac{3 z ^{5}}{4} + \frac{z ^{6}}{8}

= (\frac{9 z ^{4}}{4} + 4 z^{3} + \frac{9 z ^{2}}{2} + 3 z + 1 + \frac{3 z ^{5}}{4} + \frac{z ^{6}}{8}) (1 + z + \frac{z ^{2}}{2})

括弧を分配する

= \frac{9 z ^{4}}{4} \cdot 1 + \frac{9 z ^{4}}{4} z + \frac{9 z ^{4}}{4} \cdot \frac{z ^{2}}{2} + 4 z^{3} \cdot 1 + 4 z^{3} z + 4 z^{3} \frac{z ^{2}}{2} + \frac{9 z ^{2}}{2} \cdot 1 + \frac{9 z ^{2}}{2} z + \frac{9 z ^{2}}{2} \cdot \frac{z ^{2}}{2} + 3 z \cdot 1 + 3 zz + 3 z \frac{z ^{2}}{2} + 1 \cdot 1 + 1 \cdot z + 1 \cdot \frac{z ^{2}}{2} + \frac{3 z ^{5}}{4} \cdot 1 + \frac{3 z ^{5}}{4} z + \frac{3 z ^{5}}{4} \cdot \frac{z ^{2}}{2} + \frac{z ^{6}}{8} \cdot 1 + \frac{z ^{6}}{8} z + \frac{z ^{6}}{8} \cdot \frac{z ^{2}}{2}

簡素化

\frac{9 z ^{4}}{4} \cdot 1 + \frac{9 z ^{4}}{4} z + \frac{9 z ^{4}}{4} \cdot \frac{z ^{2}}{2} + 4 z^{3} \cdot 1 + 4 z^{3} z + 4 z^{3} \frac{z ^{2}}{2} + \frac{9 z ^{2}}{2} \cdot 1 + \frac{9 z ^{2}}{2} z + \frac{9 z ^{2}}{2} \cdot \frac{z ^{2}}{2} + 3 z \cdot 1 + 3 zz + 3 z \frac{z ^{2}}{2} + 1 \cdot 1 + 1 \cdot z + 1 \cdot \frac{z ^{2}}{2} + \frac{3 z ^{5}}{4} \cdot 1 + \frac{3 z ^{5}}{4} z + \frac{3 z ^{5}}{4} \cdot \frac{z ^{2}}{2} + \frac{z ^{6}}{8} \cdot 1 + \frac{z ^{6}}{8} z + \frac{z ^{6}}{8} \cdot \frac{z ^{2}}{2} : \frac{17 z ^{4}}{2} + 5 z^{5} + 2 z^{6} + 10 z^{3} + 8 z^{2} + 4 z + 1 + \frac{z ^{7}}{2} + \frac{z ^{8}}{16}

= \frac{17 z ^{4}}{2} + 5 z^{5} + 2 z^{6} + 10 z^{3} + 8 z^{2} + 4 z + 1 + \frac{z ^{7}}{2} + \frac{z ^{8}}{16}