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Name: scruta \(\quad\) Daily life: mowing

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1st upload: 2023/06/17
revision2 : 2023/07/27
revision3 : 2024/12/22
revision4 : 2025/09/14

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\begin{align} &f(x)=x^2+x+1 \quad v \equiv \alpha+2\beta \notag \\ \notag \\ &V(x) \equiv (x-v_{1})(x-v_{2})=g_0(x) \notag \\ &P_\alpha(x)=V(x) \bigl(\frac{\alpha }{x-{v_1}}+\frac{\beta }{x-{v_2}} \bigr) \notag \\ &\alpha=\frac{ P_\alpha(v_1)}{(v_1-v_2)} \notag \\ \notag \\ &\left\{ \begin{array}{l} h_0 = (x-v_1) \\ h_1 = (x-v_2) \\ \end{array} \right. \notag \\ &\begin{bmatrix} t_0 \\ t_1 \end{bmatrix} \equiv \frac{1}{2} \begin{bmatrix} 1&1 \\ 1&-1 \end{bmatrix} \cdot \begin{bmatrix} h_0 \\ h_1 \end{bmatrix} \notag \\ \notag \\ &\left\{ \begin{array}{l} B_0(x)=x^2-A_0=0 \\ t_1^2=A_0=-\frac{3}{4} \\ a_0=\sqrt{A_0} \ \in Q_1 \\ \end{array} \right. \notag \\ \notag \\ &\begin{bmatrix} \tilde{h_0} \\ \tilde{h_1 } \end{bmatrix} = \begin{bmatrix} 1&1 \\ 1&-1 \end{bmatrix} \cdot \begin{bmatrix} t_0 \\ \tilde{t_1} \end{bmatrix} = \begin{bmatrix} g_1(x) \\ t_0- \tilde{t_1} \end{bmatrix} \notag \\ \notag \\ &\left\{ \begin{array}{l} g_1(x)=0 \ \rightarrow \ v=-\frac{3}{2}-a_0 \\ \therefore \alpha=-v-2=-\frac{1}{2}+a_0 =\omega \end{array} \right. \notag \\ \end{align}

\begin{align} &f(x)=x^3-3x+1 \quad g(v)=v^3-9v-9 \notag \\ \notag \\ &Res(f(x+v) ,g(v),v) \notag \\ &=-(x^3-21x+37)(x^3-12x-8)(x^3-3x+1) \notag \\ \notag \\ \end{align}

\begin{align} Res&(f(x+v), g(v),v) \notag \\ =(-1)& \left\{\bbox[#FFFF00]{ (x+v_1-\alpha) }\bbox[#7FFF00]{(x+v_1-\beta)}\bbox[#00FFFF]{(x+v_1-\gamma)} \right \} \notag \\ \times &\left\{\bbox[#00FFFF]{ (x+v_4-\alpha)}\bbox[#FFFF00]{ (x+v_4-\beta)}\bbox[#7FFF00]{(x+v_4-\gamma)} \right \} \notag \\ \times &\left\{\bbox[#7FFF00]{(x+v_5-\alpha)}\bbox[#00FFFF]{(x+v_5-\beta)}\bbox[#FFFF00]{ (x+v_5-\gamma)} \right \} \notag \\ \notag \\ =(-1)& \left\{ \bbox[#FFFF00]{(x+v_1-\alpha)(x+v_4-\beta)(x+v_5-\gamma)} \right \} \notag \\ \times &\left\{\bbox[#7FFF00]{(x+v_1-\beta)(x+v_4-\gamma)(x+v_5-\alpha)} \right \} \notag \\ \times &\left\{\bbox[#00FFFF]{(x+v_1-\gamma)(x+v_4-\alpha)(x+v_5-\beta)} \right \} \notag \\ \notag \\ =(-1)& \cdot\bbox[#FFFF00]{H_1} \cdot \bbox[#7FFF00]{H_2} \cdot \bbox[#00FFFF]{H_3} \notag \\ \notag \\ \end{align}

\begin{align} &f(x+v) \notag \\ &=\bbox[#FFFF00]{(x+v-\alpha)} \cdot \bbox[#7FFF00]{(x+v-\beta)} \cdot \bbox[#00FFFF]{(x+v-\gamma)} \notag \\ \notag \\ &\left\{ \begin{array}{l} GCD\bigl(\left. \bbox[#FFFF00]{H_1} \right|_{v_1=v} \ ,f(x+v)\bigr)= \bbox[#FFFF00]{(x+v-\alpha)}=Y_1 \\ GCD\bigl(\left. \bbox[#7FFF00]{H_2} \right|_{v_1=v} \ ,f(x+v)\bigr)= \bbox[#7FFF00]{(x+v-\beta)}=Y_2 \\ GCD\bigl(\left. \bbox[#00FFFF]{H_3} \right|_{v_1=v} \ ,f(x+v)\bigr)= \bbox[#00FFFF]{(x+v+\gamma)}=Y_3 \\ \end{array} \right. \notag \\ \notag \\ \end{align}

\begin{align} f(x)&=Y_1(x-v) \cdot Y_2(x-v) \cdot Y_3(x-v) \notag \\ &=(x-x_1)(x-x_2)(x-x_3) \notag \\ \end{align}

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