Download Sample pages 1 PDF

24
2 Algebraic Identities, Equations and Systems
Proof Letting S denote the given number, we have
S D a2 .c b/ C b2 a b2 c C c2 b c2 a
D a2 .c b/ C .b2 a c2 a/ C .c2 b b2 c/
D a2 .c b/ C a.b C c/.b c/ C bc.c b/
D .c b/Œa2 a.b C c/ C bc
D .c b/.a b/.a c/;
where we used (2.3) in the last equality. Now, it follows from a ¤ b, b ¤ c and
c ¤ a that a b; c b; a c ¤ 0, so that S ¤ 0.
t
u
A useful variant of Viète’s formula is the factorisation for the expression x2 C
Sx C P, where, as before, S D y C z and P D yz:
x2 C Sx C P D .x C y/.x C z/:
(2.4)
If we change S, y and z in (2.3) respectively by S, y and z, we immediately see
that (2.4) is indeed equivalent to that factorisation.
The next example uses (2.4) to get yet another algebraic identity, which will be
further applied in a number of places, both in this volume as well as in [4] and [5].
Example 2.7 For all x; y; z 2 R, we have
.x C y C z/3 D x3 C y3 C z3 C 3.x C y/.x C z/.y C z/:
(2.5)
Proof Applying item (d) of Proposition 2.1 twice, first with x C y in place of x and
z in place of y, we successively get
.x C y C z/3 D Œ.x C y/ C z3
D .x C y/3 C z3 C 3.x C y/zŒ.x C y/ C z
D x3 C y3 C 3xy.x C y/ C z3 C 3.x C y/Œ.x C y/z C z2 /
D x3 C y3 C z3 C 3.x C y/Œxy C .x C y/z C z2 D x3 C y3 C z3 C 3.x C y/.y C z/.x C z/;
where, in the last equality, we have used the variant (2.4) of Viète’s formula.
Problems: Section 2.1
1. * Prove the other items of Proposition 2.1.
t
u