5.4 POLYNOMIALS 169Using the same reasoning, the right-hand side will have 16 monomial terms (before
collecting like terms), each formed by one choice of x or -p, x or -q, etc. For exam
ple, the terms using exactly two xs will also have exactly two constants. How many
such terms will there be?^12 The number of ways that you can pick two different el
ements from the set {p,q,r, s}; i.e., (i) = 6 terms. Working out all the terms, we
have
(x- p)(x - q)(x - r)(x - s) = x^4 - (p +q+ r + s)x^3
+(pq+ pr + ps +qr+qs+ rs)�
- (pqr+ pqs + prs + qrs)x+ pqrs.
Equating like terms, we have
a 3 = -(sum of the zeros)
a 2 = + (sum of all products of two different zeros)
a I = -(sum of all products of three different zeros)
ao = + (product of the zeros),
where it is understood that "different" here has a purely symbolic meaning; i.e. wemultiply only zeros with different labels, such as p and q, even if their numerical
values are the same.
Finally, we see the pattern, and can write the formulas in general:
Let rl , r 2 , ... ,r n be the zeros of the monic polynomial
� + an-I�-I + ... +ao = O.
Thenfor k = 1,2, ... , n,
ak
= (-It-k(sum of all products of n -k diff erent zeros)
an
= (_I)n-k � L.J ril ri 2 ... rin_k·
(^1) 5) 1 <h<··-<in-k5,n
These formulas are very important, and should be committed to memory. The
imprecise language "sum of all products ... " is easier to remember, but do take the
time to understand how the careful use of subscripts rigorously formulates the sums.
Also note the role of the power of - 1. We use the convenient fact that ( _ 1 )k is equal
to + 1 if k is even, and - 1 if k is odd.
Let's come down to earth from this abstract discussion by looking at a concrete
example.
Example 5.4.3 (USAMO 1984) The product of two of the four zeros of the quartic
equation
is - 32. Find k.
(^12) You've looked at Section 6.1, right?