170 5. Approximation and Location of Zeros
5.2 Tests for Real Zeros
How many real zeros does the polynomial t5 - 3t4 - t2 -4t + 14 have? Where
are they located? Quickly, we can say there is at least one but no more than
five. Can we find out more without having to go to the trouble of solving
the equation, even approximately? The answer is yes, there are methods
which will allow us to determine whether real roots exist and how many lie
inside a given interval without having to pin them down numerically. Three
common ones are the Descartes’ Rule of Signs, the Fourier-Budan Method
and Sturm’s Method. Each of these provides more precise information than
its predecessor, but at the expense of an increased amount of work.
In the following exercises, (a, b) refers to the open interval {x : a < x < b}
and [a, b] to the closed interval {x : a 5 x _< b}.
Exercises
- Show that the polynomial 6ts + 5t6 + 12t4 + 2t2 + 1 has no real zeros.
- Show that a nontrivial polynomial whose nonzero coefficients are all
positive reals can have no real nonnegative zeros. - Suppose that p(t) is a polynomial over R with positive leading coef-
ficient such that p(k) < 0 for some real k. Show that p(t) has a real
zero which exceeds k. - Show that the real linear polynomial at + b (ub # 0) has a positive
zero if and only if the coefficients a and b have opposite signs. - Let p(t) be a polynomial over R. Show that r is a zero of the poly-
nomial p(t) i f an d only if -r is a zero of p(-t). - Suppose a real polynomial a,$ + a,-lt”-’ +... + alt + a0 be given.
Write down its nonzero coefficients in order, and replace each positive
one by + and each negative one by -. We say that the coefficients
have k sign changes if, as we read along the sequence of + and -
signs, there are k places where there is a sign change.
(a) Verify that th e sequence of signs corresponding to 8t3 + 3t2 -t + 1
is + + - +, and that the coefficients have two sign changes.
(b) Verify that the sequence of signs corresponding to the polyno-
mial (t - 4)(8t3 + 3t2 - t + 1) is + - - + -, and that the
coefficients have three sign changes.
(c) Suppose that the coefficients of the polynomial p(t) have k sign
changes, and that r is a positive real. Show that the polynomial
(t - r)p(t) has at least k + 1 sign changes.