楼上讲的多项式长除法(Long Division)在这里并不适用,因为该三次方程并无因子(factor).
一楼所说的导数法,就是牛顿近似法(Newton Method of Approximation),比较麻烦.
本题用任何可以开立方的计算器,一次计算可以足够准确:随便估计一个数,譬如 2,2³+2×2=12,12与19相差虽然很大,但是将2代进去叠代计算时,几次叠代可以精确到小数点十位以上.
方法如下:
x³ = 19 - 2x
x.= (19 - 2x)^⅓ = (19 - 4)^⅓ = 15^⅓ = 2.4140138
叠代几次就非常准:
x₁= (19 - 2×2.4140138)^⅓ = 2.4199706
x₂= (19 - 2×2.4199706)^⅓ = 2.4192923
x₃= (19 - 2×2.4192923)^⅓ = 2.4193696
x₄= (19 - 2×2.4193696)^⅓ = 2.4193608
x5 = (19 - 2×2.4193618)^⅓ = 2.4193617
x6 = (19 - 2×2.4193617)^⅓ = 2.4193617
x7 = (19 - 2×2.4193617)^⅓ = 2.4193617
x8 = (19 - 2×2.4193617)^⅓ = 2.4193617
其他两根是虚根.