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［根与系数的关系］设

$f\left( x \right) = x^n + a_1 x^{n - 1} + \cdots + a_n$ 为复数域S上的一元多项式，x₁,x₂,…,x_n为f(x)在S中的n个根，则根与系数的关系为

$x_1 + x_2 + \cdots + x_n = \sum\limits_{i = 1}^n {x_i } = - a_1$

$x_1 x_2 + x_1 x_3 + \cdots + x_{n - 1} x_n = \sum\limits_{i,j = 1\left( {i < j} \right)}^n {x_i x_j} = a_2$

$x_1 x_2 x_3 + x_1 x_2 x_4 + \cdots + x_{n - 2} x_{n - 1} x_n = \sum\limits_{i,j,k = 1\left( {i < j < k} \right)}^n {x_i x_j x_k } =- a_3$

$\begin{array}{l} \cdots \cdots \cdots \cdots \\ x_1 x_2 \cdots x_n = \left( { - 1} \right)^n a_n \\ \end{array}$ 这就是说，f(x)的x_n-k的系数a_k等于从它的根x₁,x₂,…,x_n中每次取k个（不同的）一切可能乘积之和，若k是偶数，则取正号，若k为奇数，则取负号．

［根的范围］设ξ为复系数代数方程

$f\left( x \right) = a_0 x^n + a_1 x^{n - 1} + \cdots + a_{n - 1} x + a_n = 0 (1)$ 的根．

1°若所有系数a_i≠0(i=0,1,…,n),则|ξ|≤σ，其中为实系数代数方程

$F\left( x \right) = \left| {a_0 } \right|x^n - \left| {a_1 } \right|x^{n - 1} - \cdots - \left| {a_n } \right| = 0$ 的一个正实根．

2°设γ₁,γ₂,…,γ_n-1为任意正数，则|ξ|≤τ，其中τ为下列n个数中最大一个：

$\frac{{\left| {a_1 } \right|}}{{\left| {a_0 } \right|}} + \frac{1}{{r_1 }},\frac{{\left| {a_2 } \right|}}{{\left| {a_0 } \right|}}r_1 + \frac{1}{{r_2 }}, \cdots ,\frac{{\left| {a_{n - 1} } \right|}}{{\left| {a_0 } \right|}}r_1 r_2 \cdots r_{n - 2} + \frac{1}{{r_{n - 1} }},\frac{{\left| {a_n } \right|}}{{\left| {a_0 } \right|}}r_1 r_2 \cdots r_{n - 1}$ 特别，取γ_i=1(i=1,2,…,n－1)时，有

$\left| \xi \right| \le \max \left\{ {\frac{{\left| {a_n } \right|}}{{\left| {a_0 } \right|}},1 + \frac{{\left| {a_1 } \right|}}{{\left| {a_0 } \right|}}, \cdots ,1 + \frac{{\left| {a_{n - 1} } \right|}}{{\left| {a_0 } \right|}}} \right\}(2)$ 方程(1)中作变换x=1/y，可求出|y|的上界，因而得到

$\left| \xi \right| \ge \left( {\max \left\{ {\frac{{\left| {a_n } \right|}}{{\left| {a_n } \right|}},1 + \frac{{\left| {a_1 } \right|}}{{\left| {a_n } \right|}}, \cdots ,1 + \frac{{\left| {a_{n - 1} } \right|}}{{\left| {a_n } \right|}}} \right\}} \right)^{ - 1}(3)$ 更进一步，记(2)式右边为M，记(3)式右边为m，如果取ρ<M，使得

$\left| {a_0 } \right|\rho ^n - \left| {a_1 } \right|\rho ^{n - 1} - \left| {a_2 } \right|\rho ^{n - 2} - \cdots - \left| {a_{n - 1} } \right|\rho - \left| {a_n } \right| > 0$ 取ρ'>m，使得

$\left| {a_0 } \right|\rho '^n + \left| {a_1 } \right|\rho '^{n - 1} + \cdots + \left| {a_{n - 1} } \right|\rho ' - \left| {a_n } \right| < 0$ 那么有ρ'≤|ξ|≤ρ．

3°设γ为任意正数，则|ξ|≤τ₁，其中

$\tau _1 = \max \left\{ {\frac{1}{\gamma },\frac{{\left| {a_1 } \right|}}{{\left| {a_0 } \right|}} + \frac{{\left| {a_2 } \right|}}{{\left| {a_0 } \right|}}\gamma + \cdots + \frac{{\left| {a_1 } \right|}}{{\left| {a_0 } \right|}}\gamma ^{n - 1} } \right\}$ 特别，取γ=1，有

$\left| \xi \right| \le \max \left\{ {1,\frac{1}{{\left| {a_0 } \right|}}\sum\limits_{i = 1}^n {\left| {a_i } \right|} } \right\}$

4°若所有系数都为正实数，则

$\min \left\{ {\frac{{a_1 }}{{a_0 }},\frac{{a_2 }}{{a_1 }}, \cdots ,\frac{{a_n }}{{a_{n - 1} }}} \right\} \le \left| \xi \right| \le \max \left\{ {\frac{{a_1 }}{{a_0 }},\frac{{a_2 }}{{a_1 }}, \cdots ,\frac{{a_n }}{{a_{n - 1} }}} \right\}$

5°若方程（1）的系数满足不等式

$\left| {a_0 } \right| < \left| {a_1 } \right| - \left| {a_2 } \right| - \left| {a_3 } \right| - \cdots - \left| {a_n } \right|$ 则方程（1）至多有一个绝对值≥1的根ξ₁，而且

$\left| {\xi _1 } \right| \ge \left| {a_1 } \right| - \left| {a_2 } \right| - \left| {a_3 } \right| - \cdots - \left| {a_n } \right|$

问题？

$(d^0.5y)/dx^0.5 = sin(x-1)sin(y-1)$ == ? $(d^0.5y)/dx^0.5 -cosh(y)-sinh(y)=0$ == ? $(d^1.6y)/(dx^1.6)-int y(x) (dx)^(0.8)-y-exp(x)=0$ == ? $int y(x) (dx)^0.5 -y-exp(x)$ =0 == ? $(d^0.5y)/dx^0.5-exp(y)x=0$ == ? $(d^0.5y)/dx^0.5-exp(y)y=0$ == ? $(d^0.5y)/dx^0.5=cos(x)/xy$ == ? $y(dy^0.5)/dx^0.5-sqrt(x)-1=0$ == ? $(d^1.2y)/(dx^1.2)-2(d^0.6y)/dx^0.6+y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5=cos(y)exp(x)x$ == ? $(d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5-exp(y)sqrt(x)=0$ == ? $(d^1.6y)/(dx^1.6)-3 (d^0.8y)/dx^0.8+2y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5$ +log(y-1)-exp(x)-x=0 == ? $(d^0.5y)/dx^0.5-exp(y)sin(x)=0$ == ? $(d^0.5y)/dx^0.5 = ysin(x)/x$ == ? $y^((0.5))(x) -4 exp(x)y-exp(x)=0$ == ? $(dy^0.5)/dx^0.5 = 1/(x-y)$ == ? $dy/dx-(d^0.5y)/dx^0.5$ - y - exp(x)=0 == ? $(dy)/dx -exp(y-1)-x-x^2=0$ == ? $(d^1.2y)/(dx^1.2)-3dy^0.6/dx^0.6+2y-exp(x)=0$ == ? $dy/dx-(d^0.5y)/dx^0.5-y-1$ =0 == ? $(d^0.5y)/dx^0.5-cos(y)sin(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0$ == ? $dy/dx-exp(y-1)-exp(x)=0$ == ? $(dy)/dx - 2(d^0.5y)/dx^0.5-y-exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5 -e^(4x)-y$ =0 == ? $y^((0.5))(x) - exp(x)y-exp(x)=0$ == ? $y^((0.5))(x) - exp(x)y-4exp(x)=0$ == ? $(dy)/dx -3(d^0.5y)/dx^0.5 +2y-exp(x)=0$ == ? $y(d^0.5y)/dx^0.5-sqrt(x)-1=0$ == ? $y^((1))(x)-exp(y-1)-x=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-2y-exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0$ == ? $(d^0.5y)/dx^0.5 - log(y-1) - exp(x) + x=0$ == ? $(dy)/dx +asin(y-1) - cos(x)-x=0$ == ? $(d^1.6y)/(dx^1.6)-3(d^0.8y)/dx^0.8+2y-exp(x)=0$ == ? $(dy)/(dx) -sqrt(y-1)-x-1 =0$ == ? $(dy)/(dx) -exp(y-1)-exp(x) = 0$ == ? $(dy)/dx$ +asinh(y-1)-cosh(x)-x =0 == ? $((d^(1/2)y)/dx^(1/2))^2 -3y (dy^0.5)/dx^0.5 + 2y^2 = 0$ == ? $(dy^0.5)/dx^0.5 = cos(x)cos(y-1)$ == ? $(d^0.5y)/dx^0.5 +log(y-1)-exp(x)-x=0$ == ? $(dy^0.5)/dx^0.5 = sin(x-1)exp(y-1)$ == ? $y(d^2y)/dx^2-(dy/dx)^2+1=0$ == ? $y^((1))(x)-exp(y-1)-log(x)=0$ == ? $(d^2y)/dx^2 exp(x)- exp(y-1)=0$ == ? $(d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8-y-exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8+y-exp(x)=0$ == ? $(dy)/dx -3 (d^0.5y)/dx^0.5+2y-exp(x)=0$ == ? $y^((0.5))(x) - xy-x=0$ == ? $y(dy^3)/dx^3-x^3-3x^2-3x-1=0$ == ? $y^((1.8))(x)-2y^((0.9))(x) +y-1=0$ == ? $y^((0.5))(x)=1/(xy-1)$ == ? $y^((2))(x)y^2-x^2-2x-1=0$ == ? $((d^0.5y)/dx^0.5)^2 -5(d^0.5y)/dx^0.5 +6=0$ == ? $y^((0.5))(x) -2 exp(x)y-4exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0$ == ? $y^(0.5)(x)=2yexp(x)$ == ? $y^((0.5))(x)-exp(x)y^2=0$ == ? $(d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0$ == ? $y^((1))(x)-y^2-xy=0$ == ? $y^((1))(x)-y^((0.5))(x) -y-1=0$ == ? $y^((2))(x) -y^2-x^2=0$ == ? $y^((2))(x) -y^2-x^2-2xy=0$ == ? $y^((0.5))(x) -int y(x) (dx)^0.5-y-exp(x)=0$ == ? $d^0.5/dx^0.5 y -2cos(y)exp(x)=0$ == ? $d^0.5/dx^0.5 y -4sin(y)exp(x)=0$ == ? $(d^0.5y)/dx^0.5=sin(x^2)y$ == ? $(d^0.5y)/dx^0.5-sin(x)sin(y)=0$ == ? $(d^0.5y)/dx^0.5-sinh(x)sinh(y)=0$ == ? $y^((1))(x)=exp(x-y)-x$ == ? $x(d^0.5y)/dx^0.5-y-2x=0$ == ? $(d^0.5y)/dx^0.5=sinh(x-1)sinh(y-1)$ == ? $y^((0.5))(x)-exp(-x)y^2=0$ == ? $(d^0.5y)/dx^0.5=y/xsin(x)$ == ? $(dy)/dx-sin(x-y)-1=0$ == ? $(d^2.5y)/dx^2.5=y(d^0.5y)/dx^0.5$ == ? $(d^0.5y)/dx^0.5=y(dy)/dx$ == ? $(d^(2-i)y)/dx^(2-i)- y+x=0$ == ? $(d^2y)/dx^2=y^3x^2$ == ? $y(d^2y)/dx^2-x^2-3x-1=0$ == ? $y(d^2y)/dx^2-2x^2-3x-1=0$ == ? $(y-x-1)(d^2y)/dx^2-3x-1=0$ == ? $y^2(d^2y)/dx^2-x^2-4x-4=0$ == ? $(y-x-1)(d^2y)/dx^2-x^2-4x-4=0$ == ? $y(d^2y)/dx^2-2x^2-2x-1=0$ == ? $y(d^3y)/dx^3-6x^3-3x^2-3x-1=0$ == ? $y^((0))(x)y^((1))(x)y^((2))(x)=x^2$ == ? $y^((3))(x)y^((2))(x)=y^((1/2))(x)$ == ? $y^((3))(x)=exp(x)y^((1))(x)y^((1/2))(x)$ == ? $y^((1/2))(x)y^((3))(x)=exp(x)$ == ? $y^((1/2))(x)y^((2))(x)=exp(x)$ == ? $(d^0.5y)/dx^0.5-2xy-1=0$ == ? $y^2(d^0.5y)/dx^0.5-x^2-4x-4=0$ == ? $exp(y-1)(d^0.5y)/dx^0.5-x=0$ == ? $y(d^2y)/dx^2-(x-2)(2x-4)=0$ == ? $y(d^3y)/dx^3-6x^3-4x^2-4x-1=0$ == ? $exp(y-1)(d^2y)/dx^2-exp(x)=0$ == ? $y^2(d^2y)/dx^2-x^2-1=0$ == ? $1/y^2(d^2y)/dx^2-x^2-1=0$ == ? $(y-x-1)(d^3y)/dx^3-(x-2)(2x-4)(3x-1)=0$ == ? $(d^0.5y)/dx^0.5-2x^2y^2-8x^2=0$ == ? $(d^0.5y)/dx^0.5-2xy^2-8x=0$ == ? $(d^0.5y)/dx^0.5-y^2-2y-2=0$ == ? $(d^0.5y)/dx^0.5-log(y-1)exp(x)=0$ == ? $y(d^2y)/dx^2-(dy/dx)^2-1=0$ == ? $(d^2y)/dx^2-asin(y-1)-sin(x)-x=0$ == ? $dy/dx(x--y)-x--y-1 = 0$ == ?

根与系数的关系

问题 ？

问题？