展开 (α_{0}+α_{1}t+α_{2}t^2)^2e^{t^2}

解答

展开

(α_{0} + α_{1} t + α_{2} t^{2})^{2} e^{t^{2}}

2 α_{2} α_{0} e^{t^{2}} t^{2} + 2 α_{2} α_{1} e^{t^{2}} t^{3} + (α_{2})^{2} e^{t^{2}} t^{4} + 2 α_{1} α_{0} e^{t^{2}} t + (α_{1})^{2} e^{t^{2}} t^{2} + (α_{0})^{2} e^{t^{2}}

求解步骤

(α_{0} + α_{1} t + α_{2} t^{2})^{2} e^{t^{2}}

(α_{0} + α_{1} t + α_{2} t^{2})^{2} = 2 α_{2} α_{0} t^{2} + 2 α_{2} α_{1} t^{3} + (α_{2})^{2} t^{4} + 2 α_{1} α_{0} t + (α_{1})^{2} t^{2} + (α_{0})^{2}

= e^{t^{2}} ((α_{2})^{2} t^{4} + 2 α_{2} α_{1} t^{3} + 2 α_{2} α_{0} t^{2} + (α_{1})^{2} t^{2} + 2 α_{1} α_{0} t + (α_{0})^{2})

= e^{t^{2}} (2 α_{2} α_{0} t^{2} + 2 α_{2} α_{1} t^{3} + (α_{2})^{2} t^{4} + 2 α_{1} α_{0} t + (α_{1})^{2} t^{2} + (α_{0})^{2})

打开括号

= e^{t^{2}} \cdot 2 α_{2} α_{0} t^{2} + e^{t^{2}} \cdot 2 α_{2} α_{1} t^{3} + e^{t^{2}} (α_{2})^{2} t^{4} + e^{t^{2}} \cdot 2 α_{1} α_{0} t + e^{t^{2}} (α_{1})^{2} t^{2} + e^{t^{2}} (α_{0})^{2}

= 2 α_{2} α_{0} e^{t^{2}} t^{2} + 2 α_{2} α_{1} e^{t^{2}} t^{3} + (α_{2})^{2} e^{t^{2}} t^{4} + 2 α_{1} α_{0} e^{t^{2}} t + (α_{1})^{2} e^{t^{2}} t^{2} + (α_{0})^{2} e^{t^{2}}

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