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2.3. The Derivative 67

(c) Differentiate both sides of the equation in (b) repeatedly. Set

t = c to obtain

P(C) = co

P’(C) = Cl

p”(C) = 2!c2

and in general

p@‘(c) = 3!c3

P(~)(C) = k!ck for 0 < - k - < n.

Substitute in (b) to obtain the result.

10. What does the Taylor Expansion of p(t) about c amount to when
    c = O?


11. Form the Taylor Expansions for p(t) and q(t) about c. Multiply them
    together and verify that the result is

(P(l)(C) + (Pd’W - c) + ;(PnYkw - 4” +... *

12. (a) Use Taylor’s Theorem to establish the Binomial Expansion

(1 + t)^ = 1 + nt + (;)tz+...+( ;>tk+...+t”.

(b) Prove that

(u + b)” = a” + na”-lb +... + anskbk +... + b”.

13. Another approach to use in expanding p(t) in terms of powers of
    (t - c) is to make the substitution t = c + s, and compute p(c + s) in
    ascending powers of s. Do this for the polynomial t3 - 4t2 + 7t + 2
    and check that the coefficient of s’ is p(‘)(c)/r! for each value of r.


14. For several polynomials and values of c of your choice, find the Tay-
    lor expansion and check your answer by (i) Horner’s Method, (ii) a
    substitution of the type t = c + s, (iii) multiplying out and adding
    the terms of the expansion.


15. Multiplicity of zeros. In Exercise 2.2, we found that c is a zero of a
    polynomial p(t) i f an d only if (t - c) is a factor of p(t). Using this
    result as a basis, we can sharpen the idea of a zero.