64 2. Evaluation, Division, and Expansion(b) 2t3 + 9t2 + 8t - 4 and t2 + 5t + 6.E.21. Investigate formulae for the remainder when a polynomial f(t) is
divided by(4 (t - elk
(b) (t-al)(t-az)(t--a+-(t-a,)(c) (t - a)‘-(t - b)“.2.3 The Derivative
The algorithm given in Section 1 for expanding a given polynomial p(t) in
terms of powers of (t - c) for a constant c is a mechanical method which
does not give much insight into the structural significance of the coefficients.
We would like to be able to describe them in terms of the polynomial p(t)
and the constant c. Surprisingly, this is done through the introduction of
a concept which most students encounter in quite a different domain-
the calculus. Let us begin with two observations before continuing to the
exercises.
(1) p(t) is the sum of monomials akt k. If we have an expansion of each
monomial in terms of powers of (t - c), then the expansion of p(t) is the
sum of the expansions of the monomials.
(2) We can write p(t) = q(t)(t - c) + p(c). If we can obtain an expansion
for q(t), then we can insert it into this equation to get one for p(t). The
constant q(c) occurs as the coefficient of (t - c) in the expansion for p(t);
can we express this in terms of p? Can the coefficients of higher powers of
(t - c) be similarly identified?
Exercises
- Construct Horner’s Table for the expansion of the following polyno-
mials in terms of ascending powers of (t - c):
(4 t2
lb) t3
(c) t4.Check your result by expanding t” = [c + (t - c)]“.- Verify, by Horner’s method or otherwise, that the first two terms in
the expansion of tm in terms of powers of (t - c) are
cm + mm-‘(t - c).