This is a sub-page of a longer article, The classical Chinese version of Horner’s method: Technical considerations.
We define a ‘Wang Xiaotong cubic’ to be a function of the form
f(x) = x3 + ax2 + bx – c (1)
a, b ≥ 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) = (x–r1) (x–r2) (x–r3)
= (x–r1)(x2–(r2+r3)x +r2r3)
= x3 – (r1 + r2 + r3)x2 + (r1r2 + r1r3 + r2r3)x – r1r2r3
a = –(r1 + r2 + r3) < 0
which contradicts (1). Thus f has exactly one positive root, which was to be proved.