This is a sub-page of a longer article, The classical Chinese version of Horner’s method: Technical considerations.

Roots of a Wang Xiaotong cubic

We define a ‘Wang Xiaotong cubic’ to be a function of the form

          f(x) = x3 + ax2 + bx – c                                                          (1)
          ab ≥ 0, c > 0

and we shall prove that such a function has exactly one positive real root.

Because f is a cubic it must have at least one real root. This root cannot be zero.

Case 1. Suppose f has a negative root, –r. Then it can be factored as

        f(x) = (x+r)(x2 + βx – γ)                                                        (2)
        γ > 0

The two roots of the quadratic factor in (2) are given by

         

These are both real, and since +, one must be positive and one negative.

Case 2. If f does not have a negative root, then it must have at least one positive real root. If it has more than one positive real root, then it must have three, r1, r2, r3 > 0, so that

        f(x) = (xr1) (xr2) (xr3)

          = (xr1)(x2–(r2+r3)x +r2r3)

           = x3 – (r1 + r2 + r3)x2 + (r1r2 + r1r3 + r2r3)x – r1r2r3

whence

        a = –(r1 + r2 + r3) < 0

which contradicts (1). Thus f has exactly one positive root, which was to be proved.