# 2项式、m项式的n次方和n次根式

2项式、m项式的n次方和n次根式

中国科学院力学研究所 吴中祥

1. 推导证明2项式n次方

(x1+x2)^2=x1^2+2x1x2+x2^2

(x1^2+2x1x2+x2^2)^(1/2)= (x1+x2)

(x1+x2)^3=x1^3+3x1^2x2+3x1x2^2+x2^3

(x1^3+3x1^2x2+3x1x2^2+x2^3)^(1/3)= (x1+x2)

(x1+x2)^4=x1^4+4x1^3x2+6x1^2x2^2+4x1x2^3+x2^4

(x1^4+4x1^3x2+6x1^2x2^2+4x1x2^3+x2^4)^(1/4)= (x1+x2)

(x1+x2)^5

=x1^5+5x1^4x2+10x1^3x2^2+10x1^2x2^3+5x1x2^4+x2^5

(x1^5+5x1^4x2+10x1^3x2^2+10x1^2x2^3+5x1x2^4

+x2^5)^(1/5)= (x1+x2)

(x1+x2)^6

= x1^6+6x1^5x2+15x1^4x2^2+20x1^3x2^3+15x1^2x2^4

+6x1x2^5+x2^6

(x1^6+6x1^5x2+15x1^4x2^2+20x1^3x2^3+15x1^2x2^4

+6x1x2^5+x2^6)^(1/6)= (x1+x2)

就足以推导、证明2项式n次方

(x1+x2)^n

=x1^n+nx1^(n-1)x2+c(n2) x1^(n-2)x2^2

++c(nn-2)x1^2x2^(n-2)+c(nn-1)x1x2^(n-1)+x2^n

=x1^n+x2^n+n(x1^(n-1)x2+x1x2^(n-1))

+c(n2)(x1^(n-2)x2^2+x1^2x2^(n-2))

++c(n(n-1)/2)(x1^(n-(n-1)/2+1)x2^(n-(n-1)/2-1)

+x1^(n-(n-1)/2-1)x2^(n-(n-1)/2+1))(n为奇数)

++c(nn/2)(x1^(n-n/2)x2^(n-n/2)(n为偶数)

c(nj)=n(n-1)(n-j)/j!=c(nn+1-j)=n(n-1)(n+1-j)/(n+1-j)!

c(n1)=c(nn)=n

2.m项式的n次方

(x1+x2+x3++x(m-2)+x(m-1)+xm)^2

= x1^2+X2^2+X3^2++x(m-2)^2+x(m-1)^2+xm^2

+2[x1(x2+x3++x(m-2)+x(m-1)+xm)

+x2(x3+x4++x(m-2)+x(m-1)+xm)

+x3(x4+x5++x(m-2)+x(m-1)+xm)

++x(m-2)(x(m-1)+xm)+x(m-1)xm]

(x1+x2+x3++x(m-2)+x(m-1)+xm)^3

= x1^3+X2^3+X3^3++x(m-2)^3+x(m-1)^3+xm^3

+3[x1^2(x2+x3++x(m-2)+x(m-1)+xm)

+x2^2(x3+x4++x(m-2)+x(m-1)+xm)

+x3^2(x4+x5++x(m-2)+x(m-1)+xm)

++x(m-2)^2(x(m-1)+xm)+x(m-1)^2xm

+x1(x2^2+x3^2++x(m-2)^2+x(m-1)^2+xm^2)

+x2(x3^2+x4^2++x(m-2)^2+x(m-1)^2+xm^2)

+x3(x4^2+x5^2++x(m-2)^2+x(m-1)^2+xm^2)

++x(m-2)(x(m-1)^2+xm^2)+x(m-1)xm^2]

(x1+x2+x3++x(m-3)+x(m-2)+x(m-1)+xm)^n

=x1^n+X2^n+X3^n++x(m-2)^n+x(m-1)^n+xm^n

+n[x1^(n-1)(x2+x3++x(m-2)+x(m-1)+xm)

+x2^(n-1)(x3+x4++x(m-2)+x(m-1)+xm)

+x3^(n-1)(x4+x5++x(m-2)+x(m-1)+xm)

++x(m-2)^(n-1)(x(m-1)+xm)+x(m-1)^2xm+x(m-1)xm^2

+x1(x2^(n-1)+x3^(n-1)

++x(m-2)^(n-1)+x(m-1)^(n-1)+xm^(n-1))

+x2(x3^(n-1)+x4^(n-1)

++x(m-2)^(n-1)+x(m-1)^(n-1)+xm^(n-1))

+x3(x4^(n-1)+x5^(n-1)

++x(m-2)^(n-1)+x(m-1)^(n-1)+xm^(n-1))

++x(m-2)(x(m-1)^(n-1)+xm^(n-1))

+x(m-1)xm^(n-1)]

+C(n,2)[x1^(n-2)(x2^2+x3^2++x(m-2)^2+x(m-1)^2+xm^2)

+x2^(n-2)(x3^2+x4^2++x(m-2)^2+x(m-1)^2+xm^2)

+x3^(n-2)(x4^2+x5^2++x(m-2)^2+x(m-1)^2+xm^2)

++x(m-2)^(n-1)(x(m-1)+xm)+x(m-1)^2xm+x(m-1)xm^2

+x1^2(x2^(n-2)+x3^(n-2)

++x(m-2)^(n-2)+x(m-1)^(n-2)+xm^(n-2))

+x2^2(x3^(n-2)+x4^(n-2)

++x(m-2)^(n-2)+x(m-1)^(n-2)+xm^(n-2))

+x3^2(x4^(n-2)+x5^(n-2)

++x(m-2)^(n-2)+x(m-1)^(n-2)+xm^(n-2))

++x(m-2)^2(x(m-1)^(n-2)+xm^(n-2))

+x(m-1)^2xm^(n-2)]

++C(n,(n-1)/2)[x1^(n-(n-1)/2-1)

(x2^(n-(n-1)/2)+x3^(n-(n-1)/2)

++x(m-1)^(n-(n-1)/2)+xm^(n-(n-1)/2))

+x2^(n-(n-1)/2-1)

(x3^(n-(n-1)/2)+x4^(n-(n-1)/2)

++x(m-1)^(n-(n-1)/2)+xm^(n-(n-1)/2))

++x(m-2)^(n-(n-1)/2-1)

(x(m-1)^(n-(n-1)/2)+xm^(n-(n-1)/2))

+x(m-1)^(n-(n-1)/2-1)xm^(n-(n-1)/2)

+x1^(n-(n-1)/2)(x2^(n-(n-1)/2+1)+x3^(n-(n-1)/2+1)

++x(m-1)^(n-(n-1)/2+1)+xm^(n-(n-1)/2+1))

+x2^(n-(n-1)/2)(x3^(n-(n-1)/2+1)+x4^(n-(n-1)/2+1)

++x(m-1)^(n-(n-1)/2+1)+xm^(n-(n-1)/2+1))

++x(m-2)^(n-(n-1)/2)(x(m-1)^(n-(n-1)/2+1)

+xm^(n-(n-1)/2+1))](n为奇数)

+C(n,n/2)[x1^(n-n/2)(x2^(n-n/2)+x3^(n-n/2)

++x(m-1)^(n-n/2)+xm^(n-n/2))

+x2^(n-n/2)(x3^(n-n/2)+x4^(n-n/2)

++x(m-1)^(n-n/2)+xm^(n-n/2))

++x(m-2)^(n-n/2)(x(m-1)^(n-n/2)+xm^(n-n/2))

+x(m-1)^(n-n/2)xm^(n-n/2)(n为偶数)

2. 创新给出2项式的2种，2根式

(x1+x2)^(1/2)为复数式

(x1+x2)^(1/2)

=x1^(1/2)+x2^(1/2)+i2^(1/2)x1^(1/4)x2^(1/4)

x1^(1/2)+x2^(1/2)-i2^(1/2)x1^(1/4)x2^(1/4)

(x1^(1/2)+x2^(1/2)+i2^(1/2)x1^(1/4)x2^(1/4))

(x1^(1/2)+x2^(1/2)-i2^(1/2)x1^(1/4)x2^(1/4))

=x1+x2，是(x1+x2)^(1/2)的复数式或共轭复数式的平方，

(x1+x2)^(1/2)为数轴式

(x1+x2)^(1/2)

=(x1^(1/2)+x2^(1/2))[实数轴]

+(-1)^(1/2)2^(1/2)x1^(1/4)x2^(1/4)[(-1)^(1/2)数轴]

{(x1+x2)^(1/2)}^2

={(x1^(1/2)+x2^(1/2))[实数轴]

+(-1)^(1/2)2^(1/2)x1^(1/4)x2^(1/4)[(-1)^(1/2)数轴]}

{(x1^(1/2)+x2^(1/2))[实数轴]

+(-1)^(1/2)2^(1/2)x1^(1/4)x2^(1/4)[(-1)^(1/2)数轴]}

=(x1^(1/2)+x2^(1/2))^2-2(x1^(1/4)x2^(1/4))^2

=x1+x2

3. 创新给出2项式的3根式：

(x1+x2)^(1/3)为数轴式

(x1+x2)^(1/3)

=(x1^(1/3)+x2^(1/3))[实数轴]

+(-1)^(1/3)(3)^(1/3)

(x1^(2/9)x2^(1/9)+zx1^(1/9)x2^(2/9))[(-1)^(1/3)数轴]

验证：

((x1+x2)^(1/3))^3

=({(x1^(1/3)+x2^(1/3))[实数轴]

+(-1)^(1/3)(3)^(1/3)

(x1^(2/9)x2^(1/9)+x1^(1/9)x2^(2/9))[(-1)^(1/3)数轴]}

数轴乘

{(x1^(1/3)+x2^(1/3))[实数轴]

+(-1)^(1/3)(3)^(1/3)

(x1^(2/9)x2^(1/9)+x1^(1/9)x2^(2/9))[(-1)^(1/3)数轴]})

数轴乘

{(x1^(1/3)+x2^(1/3))[实数轴]

+(-1)^(1/3)(3)^(1/3)

(x1^(2/9)x2^(1/9)+x1^(1/9)x2^(2/9))[(-1)^(1/3)数轴]}

={(x1^(1/3)+x2^(1/3))^2[实数轴]

+[(-1)^(1/3)(3)^(1/3)

(x1^(2/9)x2^(1/9)+x1^(1/9)x2^(2/9))]^2[(-1)^(2/3)数轴]}

数轴乘

{(x1^(1/3)+x2^(1/3))[实数轴]

+(-1)^(1/3)(3)^(1/3)

(x1^(2/9)x2^(1/9)+x1^(1/9)x2^(2/9))[(-1)^(1/3)数轴]}

={(x1^(1/3)+x2^(1/3))^3

+[(-1)^(1/3)(3)^(1/3)

(x1^(2/9)x2^(1/9)+x1^(1/9)x2^(2/9))]^3}

=x1+x2

註：“数轴乘”是各“数轴”自乘成为该轴的“相应新数轴”。

4. 创新推导出2项式的n根式：

(x1+x2)^(1/n)为数轴式

(x1+x2)^(1/n)

=(x1^(1/n)+x2^(1/n))[实数轴]

+(-1)^(1/c(n,2))(c(n,2))^(1/c(n,2))

(x1^((c(n,2)-1)/n^2)x2^(c(n,2)/n^2)

+x1^(c(n,2)^2/n^2)x2^((c(n,2)+1)/n^2))

[(-1)^(1/c(n,2))数轴]

+(-1)^(1/c(n,3))(c(n,3))^(1/c(n,3))

(x1^((c(n,3)-1)/n^2)x2^(c(n,3)/n^2)

+x1^(c(n,3)/n^2)x2^((c(n,3)+1)/n^2))

[(-1)^(1/c(n,3))数轴]

+… … …+(-1)^(1/c(n,(n-1)/2))(c(n,(n-1)/2))^(1/c(n,(n-1)/2))

(x1^((c(n,(n-1)/2)-1)/n^2)x2^(c(n,(n-1)/2)/n^2)

+(x1^(c(n,(n-1)/2-1)/n^2)x2^((c(n,(n-1)/2)/n^2))

+x1^((c(n,(n-1)/2)/n^2)x2^(c(n,(n-1)/2+1)/n^2))

[(-1)^(1/c(n,(n-1)/2))数轴](n是奇数)

+… … …+(-1)^(1/c(n,n/2))(c(n,n/2))^(1/c(n,n/2))

x1^((c(n,n/2))/n^2)x2^(c(n,n/2)/n^2)

[(-1)^(1/c(n,n/2))数轴](n是偶数)

5. 创新推导出m项式的n根式：

(x1+x2++xm)^(1/n)为数轴式

(x1+x2++xm)^(1/n)

=(x1^(1/n)+x2^(1/n)++xm^(1/n))[实数轴]

+(-1)^(1/c(n,2))(c(n,2))^(1/c(n,2))

(x1^((c(n,2)-1)/n^2)

(x2^(c(n,2)/n^2)+x3^(c(n,2)/n^2)++xm^(c(n,2)/n^2))

+x1^(c(n,2)^2/n^2)

(x2^((c(n,2)+1)/n^2))+x3^((c(n,2)+1)/n^2)

++xm^((c(n,2)+1)/n^2))

[(-1)^(1/c(n,2))数轴]

+(-1)^(1/c(n,3))(c(n,3))^(1/c(n,3))

(x1^((c(n,3)-1)/n^2)

(x2^(c(n,3)/n^2)+x3^(c(n,3)/n^2)++xm^(c(n,3)/n^2))

+x1^(c(n,3)/n^2)

(x2^((c(n,3)+1)/n^2)+x3^((c(n,3)+1)/n^2)

++xm^((c(n,3)+1)/n^2))

[(-1)^(1/c(n,3))数轴]

+… … …+(-1)^(1/c(n,(n-1)/2))(c(n,(n-1)/2))^(1/c(n,(n-1)/2))

(x1^((c(n,(n-1)/2)-1)/n^2)

(x2^(c(n,(n-1)/2)/n^2)+x3^(c(n,(n-1)/2)/n^2)

++xm^(c(n,(n-1)/2)/n^2))

+(x1^(c(n,(n-1)/2)/n^2)

(x2^(c(n,(n-1)/2+1)/n^2)+x3^(c(n,(n-1)/2+1)/n^2)

++xm^(c(n,(n-1)/2+1)/n^2))

[(-1)^(1/c(n,(n-1)/2))数轴](n是奇数)

+… … …+(-1)^(1/c(n,n/2))(c(n,n/2))^(1/c(n,n/2))

x1^((c(n,n/2))/n^2)

(x2^(c(n,n/2)/n^2)+x3^(c(n,n/2)/n^2)

++xm^(c(n,n/2)/n^2))

[(-1)^(1/c(n,n/2))数轴](n是偶数)

6. 各种m项式的n次方n根式

m项式的s次方s根式的n次方n根式，就是：

(x1+x2++x(m+1))的各项顺序以m项式的s次方s根式的各项取代，的n次方n根式。

一元m次代数方程的m+1项式n次方n根式，就是：

(x1+x2++x(m+1))的各项顺序以一元m次代数方程的m+1的各项取代，的n次方n根式。

http://blog.sciencenet.cn/blog-226-1136090.html

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