簡約 ((x+h)^5-x^5)/h

解

簡約

\frac{( x + h ) ^{5} - x ^{5}}{h}

5 x^{4} + 10 x^{3} h + 10 x^{2} h^{2} + 5 x h^{3} + h^{4}

解答ステップ

\frac{( x + h ) ^{5} - x ^{5}}{h}

因数

(x + h)^{5} - x^{5} : h ((x + h)^{4} + x (x + h)^{3} + x^{2} (x + h)^{2} + x^{3} (x + h) + x^{4})

= \frac{h ( ( x + h ) ^{4} + x ( x + h ) ^{3} + x ^{2} ( x + h ) ^{2} + x ^{3} ( x + h ) + x ^{4} )}{h}

共通因数を約分する:

h

= (x + h)^{4} + x (x + h)^{3} + x^{2} (x + h)^{2} + x^{3} (x + h) + x^{4}

(x + h)^{4} : x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4}

= x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} + x (x + h)^{3} + x^{2} (x + h)^{2} + x^{3} (x + h) + x^{4}

(x + h)^{3} : x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}

= x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} + x (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) + x^{2} (x + h)^{2} + x^{3} (x + h) + x^{4}

(x + h)^{2} : x^{2} + 2 x h + h^{2}

= x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} + x (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) + x^{2} (x^{2} + 2 x h + h^{2}) + x^{3} (x + h) + x^{4}

拡張

x (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) : x^{4} + 3 x^{3} h + 3 x^{2} h^{2} + x h^{3}

= x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} + x^{4} + 3 x^{3} h + 3 x^{2} h^{2} + x h^{3} + x^{2} (x^{2} + 2 x h + h^{2}) + x^{3} (x + h) + x^{4}

拡張

x^{2} (x^{2} + 2 x h + h^{2}) : x^{4} + 2 x^{3} h + x^{2} h^{2}

= x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} + x^{4} + 3 x^{3} h + 3 x^{2} h^{2} + x h^{3} + x^{4} + 2 x^{3} h + x^{2} h^{2} + x^{3} (x + h) + x^{4}

拡張

x^{3} (x + h) : x^{4} + x^{3} h

= x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} + x^{4} + 3 x^{3} h + 3 x^{2} h^{2} + x h^{3} + x^{4} + 2 x^{3} h + x^{2} h^{2} + x^{4} + x^{3} h + x^{4}

簡素化

x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4} + x^{4} + 3 x^{3} h + 3 x^{2} h^{2} + x h^{3} + x^{4} + 2 x^{3} h + x^{2} h^{2} + x^{4} + x^{3} h + x^{4} : 5 x^{4} + 10 x^{3} h + 10 x^{2} h^{2} + 5 x h^{3} + h^{4}

= 5 x^{4} + 10 x^{3} h + 10 x^{2} h^{2} + 5 x h^{3} + h^{4}