68 2. Evaluation, Division, and Expansion(a) Show that, for any number c and any nonzero polynomial p(t),
we can find a nonnegative integer r, not exceeding the degree of
p(t), such that
p(t) = (t - c)‘q(t)
where q(t) is a polynomial with q(c) # 0. Thus, (t - c)’ di-
vides p(t) while (t - c) r+l does not. The number r is called the
multiplicity of c as a zero of p(t). If c is not a zero then c has
multiplicity 0. A zero of multiplicity one is called a simple zero.
(b) Let p(c) = 0. Sh ow that the multiplicity of c as a zero of p(t)
exceeds 1 if and only if c is a zero of p’(t). In this situation, show
that the multiplicity of c as a zero of p(t) exceeds the multiplicity
of c as a zero of p’(t) by 1.
(c) Show that (t - c ) is a zero of positive multiplicity r if and only
ifp(c)=p’(c)=...= p(‘-‘j(c) = 0 and p(‘)(c) # 0.
(d) What is the term of lowest degree in the Taylor expansion of
p(t) about c if c is a zero of multiplicity r?- Let p be a polynomial and c a constant. Prove or disprove: if p(c) =
p”(c) = 0, then c is a zero of p of multiplicity at least three.
Explorations
E.22. Higher Order Derivatives of the Composition of Two Func-
tions. Let p and q be any two polynomials. Is there a general formula
for the kth derivative (p o q)(‘)(t) of their composition? To deal with this
question, let us introduce some notation:
pk to denote p(l)(q(t)), the kth derivative of p evaluated at q(t);
qk to denote p’(t), the kth derivative of q at t.
Verify that
(P 0 d’(t) = PlQl
(P 0 q)“(t) = PZQ? + Pl Q2
(P 0 !I)“‘@) = P3Qf + 3PzQ142 + PlQ3 *
Compute derivatives of the next few higher orders and look for patterns.
For example, try to get the profile of a general term without regard to the
exact value of the coefficient; relate the subscript for p to the powers of the
derivatives of q in each product (can you explain the relationship). What
is the nature and the value of the coefficient of the term with the factor
pk-1 in the development of the kth derivative?
E.23. Partial Derivatives. The ring F[z, y], of polynomials in two vari-
ables over a field F can be thought of in two ways:
(i) as a ring of polynomials in the variable z over F[y];