118196 If \(\alpha\) and \(\beta\) are two complex roots of the equation
\(6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0 \text {, then } \alpha+\beta\)
\(=\)

118197 If \(\omega\) is a non-real cube root of unity and \(x=\omega^2-\omega-3\), then the value of \(x^4+6 x^3+10 x^2-12 x-19\) is

118198 For the equation \(x^4+x^3-4 x^2+x-1=0\) the ratio of the sum of the squares of all the roots to the product of the distinct roots is

118199 If \(\alpha_1, \beta_1, \gamma_1, \delta_1\) are the roots of the equation \(a x^4+b x^3+c x^2+d x+e=0\) and \(\alpha_2, \beta_2, \gamma_2, \delta_2\) are the roots of the equation
\(\mathbf{e x}^4+\mathbf{d x} x^3+\mathbf{c x}^2+\mathbf{b x}+\mathbf{a}=\mathbf{0} \text { such that }\)
\(\mathbf{0}\lt \boldsymbol{\alpha}_1\lt \boldsymbol{\beta}_1\lt \gamma_1\lt \boldsymbol{\delta}_1, \boldsymbol{0}\lt \boldsymbol{\alpha}_2\lt \boldsymbol{\beta}_2\lt \gamma_2\lt \boldsymbol{\delta}_2,\)
\(\boldsymbol{\alpha}_1-\boldsymbol{\delta}_2=\mathbf{2}=\boldsymbol{\beta}_1-\gamma_2 ; \gamma_1-\boldsymbol{\beta}_2=\boldsymbol{\delta}_1-\boldsymbol{\alpha}_2=\mathbf{4} \text {, then }\)
\(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}+\mathbf{e}=\)

118196 If \(\alpha\) and \(\beta\) are two complex roots of the equation
\(6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0 \text {, then } \alpha+\beta\)
\(=\)

118197 If \(\omega\) is a non-real cube root of unity and \(x=\omega^2-\omega-3\), then the value of \(x^4+6 x^3+10 x^2-12 x-19\) is

118198 For the equation \(x^4+x^3-4 x^2+x-1=0\) the ratio of the sum of the squares of all the roots to the product of the distinct roots is

118199 If \(\alpha_1, \beta_1, \gamma_1, \delta_1\) are the roots of the equation \(a x^4+b x^3+c x^2+d x+e=0\) and \(\alpha_2, \beta_2, \gamma_2, \delta_2\) are the roots of the equation
\(\mathbf{e x}^4+\mathbf{d x} x^3+\mathbf{c x}^2+\mathbf{b x}+\mathbf{a}=\mathbf{0} \text { such that }\)
\(\mathbf{0}\lt \boldsymbol{\alpha}_1\lt \boldsymbol{\beta}_1\lt \gamma_1\lt \boldsymbol{\delta}_1, \boldsymbol{0}\lt \boldsymbol{\alpha}_2\lt \boldsymbol{\beta}_2\lt \gamma_2\lt \boldsymbol{\delta}_2,\)
\(\boldsymbol{\alpha}_1-\boldsymbol{\delta}_2=\mathbf{2}=\boldsymbol{\beta}_1-\gamma_2 ; \gamma_1-\boldsymbol{\beta}_2=\boldsymbol{\delta}_1-\boldsymbol{\alpha}_2=\mathbf{4} \text {, then }\)
\(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}+\mathbf{e}=\)

118196 If \(\alpha\) and \(\beta\) are two complex roots of the equation
\(6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0 \text {, then } \alpha+\beta\)
\(=\)

118197 If \(\omega\) is a non-real cube root of unity and \(x=\omega^2-\omega-3\), then the value of \(x^4+6 x^3+10 x^2-12 x-19\) is

118198 For the equation \(x^4+x^3-4 x^2+x-1=0\) the ratio of the sum of the squares of all the roots to the product of the distinct roots is

118199 If \(\alpha_1, \beta_1, \gamma_1, \delta_1\) are the roots of the equation \(a x^4+b x^3+c x^2+d x+e=0\) and \(\alpha_2, \beta_2, \gamma_2, \delta_2\) are the roots of the equation
\(\mathbf{e x}^4+\mathbf{d x} x^3+\mathbf{c x}^2+\mathbf{b x}+\mathbf{a}=\mathbf{0} \text { such that }\)
\(\mathbf{0}\lt \boldsymbol{\alpha}_1\lt \boldsymbol{\beta}_1\lt \gamma_1\lt \boldsymbol{\delta}_1, \boldsymbol{0}\lt \boldsymbol{\alpha}_2\lt \boldsymbol{\beta}_2\lt \gamma_2\lt \boldsymbol{\delta}_2,\)
\(\boldsymbol{\alpha}_1-\boldsymbol{\delta}_2=\mathbf{2}=\boldsymbol{\beta}_1-\gamma_2 ; \gamma_1-\boldsymbol{\beta}_2=\boldsymbol{\delta}_1-\boldsymbol{\alpha}_2=\mathbf{4} \text {, then }\)
\(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}+\mathbf{e}=\)

118196 If \(\alpha\) and \(\beta\) are two complex roots of the equation
\(6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0 \text {, then } \alpha+\beta\)
\(=\)

118197 If \(\omega\) is a non-real cube root of unity and \(x=\omega^2-\omega-3\), then the value of \(x^4+6 x^3+10 x^2-12 x-19\) is

118198 For the equation \(x^4+x^3-4 x^2+x-1=0\) the ratio of the sum of the squares of all the roots to the product of the distinct roots is

118199 If \(\alpha_1, \beta_1, \gamma_1, \delta_1\) are the roots of the equation \(a x^4+b x^3+c x^2+d x+e=0\) and \(\alpha_2, \beta_2, \gamma_2, \delta_2\) are the roots of the equation
\(\mathbf{e x}^4+\mathbf{d x} x^3+\mathbf{c x}^2+\mathbf{b x}+\mathbf{a}=\mathbf{0} \text { such that }\)
\(\mathbf{0}\lt \boldsymbol{\alpha}_1\lt \boldsymbol{\beta}_1\lt \gamma_1\lt \boldsymbol{\delta}_1, \boldsymbol{0}\lt \boldsymbol{\alpha}_2\lt \boldsymbol{\beta}_2\lt \gamma_2\lt \boldsymbol{\delta}_2,\)
\(\boldsymbol{\alpha}_1-\boldsymbol{\delta}_2=\mathbf{2}=\boldsymbol{\beta}_1-\gamma_2 ; \gamma_1-\boldsymbol{\beta}_2=\boldsymbol{\delta}_1-\boldsymbol{\alpha}_2=\mathbf{4} \text {, then }\)
\(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}+\mathbf{e}=\)