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根与系数的关系

[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,…,xnf(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xn-k的系数ak等于从它的根x1,x2,…,xn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ai≠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,…,γn-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γi=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 \[ \left| {\xi _1 } \right| \ge \left| {a_1 } \right| - \left| {a_2 } \right| - \left| {a_3 } \right| - \cdots - \left| {a_n } \right| \]



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参阅
  1. 数学 - 数学符号 - 数学索引
  2. 手册 = 初中数学手册 + 高中数学手册 + 数学手册 + 实用数学手册
  3. 初等数学 = 小学数学 + 中学数学 ( 初中数学 + 高中数学 )
  4. 高等数学 = 基础数学 ( 代数 + 几何 + 分析 ) + 应用数学
  5. 公式 - 定理 - - 函数图 - 曲线图 - 平面图 - 立体图 - 动画 - 画画
  6. 书单 = 数学 + 物理 + 化学 + 计算 + 医学 + 英语 + 教材 - QQ群下载书
  7. 数学手册计算器 = 数学 + 手册 + 计算器 + 计算机代数系统
  8. 检测 - 例题 :

`(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)/x*y` == ? `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 = y*sin(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) - x*y-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/(x*y-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)=2y*exp(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-x*y=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^2-2x*y=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/x*sin(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^3*x^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^0.5-2x*y-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^2*y^2-8x^2=0` == ? `(d^0.5y)/dx^0.5-2x*y^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` == ?


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