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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
提 要
推导证明了2项式、m项式,的n次方,创新论证给出了2项式、m项式,的n次根式,特别是,已证明普遍存在正、负数的高次根式,为有关研究发展提供了必要的重要工具。
关键词:2项式,多项式,乘方,根式,
1. 推导证明2项式n次方
逐次对2项式,具体算得其2至6次方,就可以推导证明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(n,2) x1^(n-2)x2^2
+…+c(n,n-2)x1^2x2^(n-2)+c(n,n-1)x1x2^(n-1)+x2^n
=x1^n+x2^n+n(x1^(n-1)x2+x1x2^(n-1))
+c(n,2)(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)),(当n为奇数)
+…+c(n,n/2)(x1^(n-n/2)x2^(n-n/2),(当n为偶数)
其中,c(n,j);j=2,3,…,n-1,是从n个中取j个的组合数,有:
c(n,j)=n(n-1)…(n-j)/j!=c(n,n+1-j)=n(n-1)…(n+1-j)/(n+1-j)!,
c(n,1)=c(n,n)=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))],(当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),(当n为偶数)
2. 创新给出2项式的2种,2次根式:
设(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),
其共轭复数式为:
x1^(1/2)+x2^(1/2)-i2^(1/2)x1^(1/4)x2^(1/4),
验证:
(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,
设(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],
验证:
{(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]}
数轴乘
{(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次根式:
设(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次根式:
(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],(当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],(当n是偶数)
5. 创新推导出m项式的n次根式:
(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],(当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],(当n是偶数)
6. 各种m项式的n次方或n次根式
m项式的s次方或s次根式的n次方或n次根式,就是:
将(x1+x2+…+x(m+1))的各项顺序以m项式的s次方或s次根式的各项取代,的n次方或n次根式。
一元m次代数方程的m+1项式的n次方或n次根式,就是:
将(x1+x2+…+x(m+1))的各项顺序以一元m次代数方程的m+1项的各项取代,的n次方或n次根式。
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