展開する (x_{0}+αx^{1/2})^3

解

展開する

(x_{0} + α x^{\frac{1}{2}})^{3}

x_{0}^{3} + 3 x_{0}^{2} α x^{\frac{1}{2}} + 3 x_{0} α^{2} x + α^{3} x^{\frac{3}{2}}

解答ステップ

(x_{0} + α x^{\frac{1}{2}})^{3}

完全立方式を適用する:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(x_{0} + α x^{\frac{1}{2}})^{3} = x_{0}^{3} + 3 x_{0}^{2} α x^{\frac{1}{2}} + 3 x_{0} (α x^{\frac{1}{2}})^{2} + (α x^{\frac{1}{2}})^{3}

= x_{0}^{3} + 3 x_{0}^{2} α x^{\frac{1}{2}} + 3 x_{0} (α x^{\frac{1}{2}})^{2} + (α x^{\frac{1}{2}})^{3}

(α x^{\frac{1}{2}})^{2} = α^{2} x

(α x^{\frac{1}{2}})^{3} = α^{3} x^{\frac{3}{2}}

= x_{0}^{3} + 3 x_{0}^{2} α x^{\frac{1}{2}} + 3 x_{0} α^{2} x + α^{3} x^{\frac{3}{2}}