2.3 METHODS OF ARGUMENT 43has at least one real root. Let us assume that the conclusion is false, i.e., that Q(x) has
no real roots. Thus Q(x) is always positive or always negative for all real x. Without
loss of generality, assume that Q(x) > 0 for all real x, in which case a > O.
Now our strategy is to use the inequality involving Q to produce a contradiction,
presumably by using the hypothesis about P in some way. How are the two polynomi
als related? We can write
Q(x) = ax^4 +bx^3 +cx^2 +dx+e
= ax^4 + (c + b )x^2 + (e + d) + bx^3 - b^2 + dx -d,
hence
Q(x) =P(^2 )+(x -l )(bx^2 +d). (6)
Now let Y be a root of P. By hypothesis, y > (^1). Consequently, if we set u := ..;Y,
we have u > 1 and P(u^2 ) = O. Substituting x = u into (6) yields
Q(u) = P (u^2 ) + (u - 1 )(bu^2 +d) = (u - 1)(bu^2 +d).
Recall that we assumed Q always to be positive, so (u - 1) (bu^2 + d) > O. But we can
also plug in x = -u into (6) , and we get
Q( -u) = P(u^2 ) + (-u - 1 )(bu^2 +d) = (-u - 1 )(bu^2 +d).
So now we must have both (u - 1)(b u^2 +d) > 0 and (-u - 1)(b u^2 +d) > O. Butthis is
impossible, since u - 1 and -u - 1 are respectively positive and negative (remember,
u > 1). We have achieved our contradiction, so our original assumption that Q was
always positive has to be false. We conclude that Q must have at least one real root. _
Why did contradiction work in this example? Certainly, there are other ways to
prove that a polynomial has at least one real root. What helped us in this problem
was the fact that the negation of the conclusion produced something that was easy to
work with. Once we assumed that Q had no real roots, we had a nice inequality that
we could play with fruitfully. When you begin thinking about a problem, it is always
worth asking,
What happens if we negate the conclusion? Will we have something
that we can work with easily?
If the answer is "yes," then try arguing by contradiction. It won't always work, but
that is the nature of investigation. To return to our old mountaineering analogy, we
are trying to climb. Sometimes the conclusion seems like a vertical glass wall, but
its negation has lots of footholds. Then the negation is easier to investigate than the
conclusion. It's all part of the same underlying opportunistic strategic principle:
Anything that fu rthers your investigation is worth doing.
The next example involves some basic number theory, a topic that we develop in
more detail in Chapter 7. However, it is important to learn at least a minimal amount
of "basic survival" number theory as soon as possible. We will discuss several number