展開する (1-x+x^2)^4

解

展開する

(1 - x + x^{2})^{4}

x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 19 x^{4} - 16 x^{3} + 10 x^{2} - 4 x + 1

解答ステップ

(1 - x + x^{2})^{4}

簡素化

(1 - x + x^{2})^{4} : (1 - x + x^{2}) (1 - x + x^{2}) (1 - x + x^{2}) (1 - x + x^{2})

= (1 - x + x^{2}) (1 - x + x^{2}) (1 - x + x^{2}) (1 - x + x^{2})

拡張

(1 - x + x^{2}) (1 - x + x^{2}) : x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1

= (x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1) (1 - x + x^{2}) (1 - x + x^{2})

拡張

(x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1) (1 - x + x^{2}) : x^{6} - 3 x^{5} + 6 x^{4} - 7 x^{3} + 6 x^{2} - 3 x + 1

= (x^{6} - 3 x^{5} + 6 x^{4} - 7 x^{3} + 6 x^{2} - 3 x + 1) (1 - x + x^{2})

括弧を分配する

= x^{6} \cdot 1 + x^{6} (- x) + x^{6} x^{2} - 3 x^{5} \cdot 1 - 3 x^{5} (- x) - 3 x^{5} x^{2} + 6 x^{4} \cdot 1 + 6 x^{4} (- x) + 6 x^{4} x^{2} - 7 x^{3} \cdot 1 - 7 x^{3} (- x) - 7 x^{3} x^{2} + 6 x^{2} \cdot 1 + 6 x^{2} (- x) + 6 x^{2} x^{2} - 3 x \cdot 1 - 3 x (- x) - 3 x x^{2} + 1 \cdot 1 + 1 \cdot (- x) + 1 \cdot x^{2}

簡素化

x^{6} \cdot 1 + x^{6} (- x) + x^{6} x^{2} - 3 x^{5} \cdot 1 - 3 x^{5} (- x) - 3 x^{5} x^{2} + 6 x^{4} \cdot 1 + 6 x^{4} (- x) + 6 x^{4} x^{2} - 7 x^{3} \cdot 1 - 7 x^{3} (- x) - 7 x^{3} x^{2} + 6 x^{2} \cdot 1 + 6 x^{2} (- x) + 6 x^{2} x^{2} - 3 x \cdot 1 - 3 x (- x) - 3 x x^{2} + 1 \cdot 1 + 1 \cdot (- x) + 1 \cdot x^{2} : x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 19 x^{4} - 16 x^{3} + 10 x^{2} - 4 x + 1

= x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 19 x^{4} - 16 x^{3} + 10 x^{2} - 4 x + 1