5.2. Tests for Real Zeros 173
(a) If a # band q(a) = q(b), s h ow f rom Rolle’s Theorem that p(t) =
q’(t) has a zero in the interval (a,b).
(b) Let q(t) have m real zeros counting multiplicity. Show that p(t)
has at least m - 1 real zeros counting multiplicity.
(c) Let p(t) = 2t3 - 3t2 - t + 1. Determine a polynomial q(t) for
which q’(t) = p(t). If possible, factor q(t) and use the result to
determine intervals which contain the zeros of p(t).
(4 Let ~0, al,... , a, be real numbers satisfying as + ai/2 + ~$3 +.. .+a,/(n+l) = 0. Show that the equation ao+alt+.. .+a,t” =
0 has at least one real root.
16. Let f(t) be any polynomial over R of degree exceeding 1. Prove that
there is a real value of H for which not all the zeros of f(t) + k are
real.
17. Theorem of Fourier and Budan. A more sophisticated test than
Descartes’ Rule of Signs is given by the following. Suppose p(t) is
a polynomial over R, and that 11 and v are reals with ‘u < v and
P(U)P(V) # 0. Th e number of zeros between u and v cannot be greater
than A-B, where A is the number of changes of sign in the sequence
(p(u),p’(‘u), p”(u),.. .) and B is th e number of changes of sign in the
sequence (p(v),p’(v),p”(v),.. .). If this number differs from A - B, it
must do so by an even amount.
(a) Let p(t) = t5-t4-t3+4t2 -t- 1. Form the sequences (p(u),p’(u),
p”(u),.. .) for u = -2, -1, 0, 1. Verify that the signs of the terms
of the sequences are given in the following table:u = -2
u = -1; I;+-+-+) 5changes
- +) 4 changes
‘u= 0 : (--+--+) 3changes
- +) 4 changes
u= (^1) : (++++++) Ochanges
Deduce from the Fourier-Budan Theorem that there is exactly
one real zero in each of the intervals (-2, -1) and (-1, 0), and
either one or three real zeros in the interval (0,l).
(b) Use the Fourier-Budan Theorem to verify that 8t2 - 8t + 1 has
one zero in each of the intervals (0,1/2) and (l/2,1).
(c) Obtain Descartes’ Rule of Signs for positive zeros as a conse-
quence of the Fourier-Budan Theorem.
(d) Show that for the polynomial t4 + t2 + 4t - 3, Descartes’ Rule of
Signs gives a sharper estimate for the number of negative zeros
than the Fourier-Budan Theorem.
- Verify the Fourier-Budan Theorem for the following quadratics by
computing their roots: