170 CHAPTER 5 ALGEBRA
Solution: Let the zeros be a, b, e, d. Then the relationship between zeros and
coefficients yieldsa+b+e+d = 18,
ab+ae+ad+be+bd+ed = k,
abc + abd + aed + bed = -200,
abed = - 1984.
Without loss of generality, let ab = - 32. Substituting this into abed = -1984 yields
cd = 62, and substituting this in tum yields the system
a+b+e+d = 18
30+ae+ad+be+bd = k
-32e - 32d +62a + 62b = -200
(3)(4)
(5)Let us think strategically. We need to compute k, not the values a,b,e,d. A
penultimate step is evaluating ae + ad + be + bd. Notice that this factors:
ae+ad +be+bd = a(e+d) +b(e+d) = (a+b)(e+d).
While we're at it, let's factor (5) as well:-32(e+d) +62(a +b) = - 200.
Now it should be clear how to proceed. We need only find the two values u := a + b
and v := e + d. Equations (3) and (5) become the system
u+v= 18,
62u - 32v = - 200,
which can be easily solved (u = 4, v = 14). Finally, we have
k = 30 + 4. 14 = 86.
Rational Roots TheoremSuppose that P(x) E Z[x] has the zero x = 2/3. Does this give you any information
about P(x)? By the Factor Theorem,
P(x) = (x -�) Q(x),
where Q(x) is a polynomial. But what kind of coefficients does Q(x) have? All that we
know for sure are that the coefficients must be rational. However, if x -� is a factor,