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*) = *x*^{3} + *ax*^{2} + *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*)(*x*^{2} + *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, *r*_{1}, *r*_{2}, *r*_{3} > 0, so that

* f*(*x*) = (*x*–*r*_{1}) (*x*–*r*_{2}) (*x*–*r*_{3})

= (*x*–*r*_{1})(*x*^{2}–(*r*_{2}+*r*_{3})*x* +*r*_{2}*r*_{3})

= *x*^{3} – (*r*_{1} + *r*_{2} + *r*_{3})*x*^{2} + (*r*_{1}*r*_{2} +* r*_{1}*r*_{3} +* r*_{2}*r*3)*x* –* r*_{1}*r*_{2}*r*_{3}

whence

* a* = –(*r*_{1} + *r*_{2} + *r*_{3}) < 0

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