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多项式的分解,余数定理,综合除法

[多项式的分解]设f(x)为实数域上的多项式,若有非常数的实系数多项式g(x)和h(x),使得 \[ f\left( x \right) = g\left( x \right)h\left( x \right) \] 则称f(x)为实数域上可约(或可化),否则称f(x)为实数域上的不可约多项式.

2°实数域上不可约多项式,除一次多项式外,只有含(共轭)复根的二次多项式.

3°每个实系数多项式都可分解为实系数的一次因式与二次因式之积.

[余数定理与综合除法]c为一常数,则多项式f(x)除以(xc)所得的余数等于f(c).设 \[ f\left( x \right) = a_0 x^n + a_1 x^{n - 1} + \cdots + a_{n - 1} x + a_n \] 求f(x)除以(xc)的商式与余数其计算格式如下: \[ \begin{array}{*{20}c} {\left. c \right)} & {a_0 } & {a_1 } & {a_2 } & \cdots & {a_{n - 1} } & {a_n } \\ {} & {} & {b_0 c} & {b_1 c} & \cdots & {b_{n - 2} } & {b_{n - 1} c} \\ \hline {} & {b_0 } & {b_1 } & {b_2 } & \cdots & {b_{n - 1} } & {b_n } \\ \end{array} \] 式中b0=a0,bi=ai+bi-1c(i=1,2,…,n) 于是得到商式\[ q\left( x \right) = b_0 x^{n - 1} + b_1 x^{n - 2} + \cdots + b_{n - 1} \] 余数 \[ r = b_n = f\left( c \right) \]



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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(4x)*y-exp(4x)=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)*(x^2-2x*y+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)*(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` == ? `2*x/y-3*y^2/x^4+(-x^2/y^2+1/y^(1/2)+2*y/x^3)*dy/dx = 0` == ?


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