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48 1. Fundamentals

9.12. Let the given polynomial be equal to c[fi(~,y)]~. Since constant
terms of the fi must vanish, we can write fi(x, y) = eiZ2 + bixy +
ciy2 + uix + viy. Use vectors: let a = (or, ~2, c13), etc. and verify that
a.b=a-u=b~c=c~v=u~v=O,b~u+a~v=O,b-v+u.c=O.
Try u = (0, l,O), v = (O,O, 1).

9.13. Compare F,, and (x+Y)~F,,-~ with a view to setting up an induction
process. Note that (x + Y)~ = & + xy.

9.14. The hard part is to show symmetry in c and I. Establish this by
induction. Look at Pn+l(x, y, z) - P,,+l(z, y, x).