一元三次方程的韦达定理的证明?
来源:学生作业帮 编辑:搜搜做题作业网作业帮 分类:数学作业 时间:2024/08/17 18:52:27
一元三次方程的韦达定理的证明?
![一元三次方程的韦达定理的证明?](/uploads/image/z/19987131-3-1.jpg?t=%E4%B8%80%E5%85%83%E4%B8%89%E6%AC%A1%E6%96%B9%E7%A8%8B%E7%9A%84%E9%9F%A6%E8%BE%BE%E5%AE%9A%E7%90%86%E7%9A%84%E8%AF%81%E6%98%8E%3F)
ax^3+bx^2+cx+d
=a(x-x1)(x-x2)(x-x3)
=a[x^3-(x1+x2+x3)x^2+(x1x2+x2x3+x1x3)x-x1x2x3]
对比系数得
-a(x1+x2+x3)=b
a(x1x2+x2x3+x1x3)=c
a(-x1x2x3)=d
即得
x1+x2+x3=-b/a
x1x2+x2x3+x1x3=c/a
x1x2x3=-d/a
=a(x-x1)(x-x2)(x-x3)
=a[x^3-(x1+x2+x3)x^2+(x1x2+x2x3+x1x3)x-x1x2x3]
对比系数得
-a(x1+x2+x3)=b
a(x1x2+x2x3+x1x3)=c
a(-x1x2x3)=d
即得
x1+x2+x3=-b/a
x1x2+x2x3+x1x3=c/a
x1x2x3=-d/a