Unknown

Notes on Explorations 405

the coefficient of x’ in the expansion of (1 + x)“, from

Horner’s table we find that

...

It can be shown by induction on r that is equal to

n(n - l)(n - 2)... (n - r + 1)

r. I

This can be handled by summation techniques discussed in Exploration 18.

E.18. The standard reference for finite differences is L.M. Milne-Thomson,
The Calculus of Finite Diflerences, (Macmillan, London, 1933). For a lighter
treatment, consult H. Freeman, Mathematics for Actuarial Students, Part
II (Cambridge, 1952).

E.19. For an introduction to coloring problems with some key references,
see W.T. Tutte, Chromials, Studies in Graph Theory, Part II (ed. D.R.
Fulkerson), (Studies in Mathematics, MAA, 1975), p. 361-377. Also, see
Chapter 9 of C.L. Liu, Introduction 20 Combinatorial Mathematics,
(McGraw-Hill, 1968) and Chapter 8 of Alan Tucker, Applied Combina-
torics, (Wiley, 1980). The chromatic polynomials for the five Platonic solids
are discussed in D.H. Lehmer, Coloring the Platonic solids, Amer. Math.
Monthly 93 (1986), 288-292.

E.20. A survey of the techniques of factoring polynomials is given in Section
4.6.2 (pages 420-441) of Donald E. Knuth, The Art of Computer Progmm-
ming, Vol. 2: Semi-Numerical Algorithms, (Addison-Wesley). The greatest
common divisor is discussed on pages 434-436.

E.21. The remainder for division by (t - c)~ is conveniently provided by
Taylor’s Theorem which renders the polynomial in the form

q(t)@ -c)k +ak-&-c)k-’ +“‘+al(t- C)+aO.