epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

54 The Nuts and Bolts of Proof, Third Edition
2. Inductive hypothesis. Assume the statement is true for all polynomials
with real coefficients whose degree is between 2 and an arbitrary
number /c, including k.
3. Deductive proof. Is the statement true for polynomials with real coefficients
of degree fe + 1? Let Q{x) = au+ix'"'^^ + UkX^ H h aix + ao
be one of such polynomials.
If Q{x) has no real zeroes, then the statement is true.
Assume that Q(x) has at least one real zero, c, which could be a
repeated zero.
Then using the rules of algebra we can write:
Q(x) =
(x-cyT(x)
where t > 1 and T{x) is a polynomial of degree (fe +1) — t, with real
coefficients.
If r=z/c+ 1, then T{x) is a constant, and the statement is true because
Q{x) has exactly fc + 1 coincident zeroes.
If t < fc + 1, then T{x) is a polynomial of degree between 1 and /c, with
real coefficients. So, by the inductive hypothesis and the base case, T{x)
has at most (fc4-1) — t real zeroes.
As every zero of T{x) is a zero of Q{x), as well, all the zeroes of T{x)
must be counted as zeroes of Q{x).
Thus, the real zeroes of Q{x) are all the real zeroes of T{x) and c, which
might be counted as a zero exactly t times. So, Q{x) has at most
{[(k + 1) - r] + t} = fc + 1 real zeroes.
Therefore, by the strong principle of mathematical induction, the statement
is true for all n> 1. •
One important comment regarding this kind of proof (by mathematical
induction) is that the inductive hypothesis must be used in the construction
of the proof of the last step. If this does not happen, then we are either using
the wrong technique or making a mistake in the construction of the
deductive step. An illustration of this is given in the next example.
EXAMPLE 5. The difference of powers 7^—4^ is divisible by 3 for all fc > 1.
Incorrect Use of Proof by Induction
1. Base case. The statement is true for the smallest number we can
consider, which is 1, because 7^—4^ = 3.
2. Inductive hypothesis. Assume the statement is true for an arbitrary
number n; that is, 7"-4" = 3t for an integer number t.
3. Deductive proof Is the statement true for n +1? Is 7"+^-4"+^ = 3s
for an integer number s?