The Art and Craft of Problem Solving

170 CHAPTER 5 ALGEBRA

Solution: Let the zeros be a, b, e, d. Then the relationship between zeros and

coefficients yields

a+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 Theorem

Suppose 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,

then 3 (x - �) = 3x - 2 will also be a factor, so we can write

P(x) = (3x - 2)S(x),

where S(x) = Q(x)/3. We know that P(x) has integer coefficients; can we say the

same thing about S(x)? Indeed we can; this is Gauss's Lemma: