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52 2. Evaluation, Division, and Expansion

(c) Construct the Horner table for evaluating at t = 3 the polyno-

mial

t3 + 3t2 + lot + 27.

(d) Use the table in (c) to express the polynomial there in the form

q(t)(t - 3) + r, for some polynomial q(t) and constant r. Check

your answer by direct computation.

(e) Write q(t) in the form u(t)(t - 3) + v for some polynomial u(t)

and constant v.

(f) Combine the results of the previous parts of this problem to

write the polynomial t4 + t2 - 3t + 7 in the form

b. + bl(t - 3) + b2(t - 3)2 + b,(t - 3)3 + b4(t - 3)4

for some constants bi. Show how the computation can be dis-

played in a convenient table.

9. Explain the connection between the table

3 -2 4 7

31 5

3 1 5 12

3 4

(^3 4 9)
3
3 7
3
and the identity
3t3 - 2t2 + 4t + 7 = 12 + 9(t - 1) + 7(t - 1)2 + 3(t - 1)s.

10. For each of the following polynomialsp(t) and constants c, use Horner’s
    Method to write p(t) in the form (t - c)q(t) +p(c) and expand p(t) in
    terms of powers of (t - c). In each case, check your answer by making
    the substitution t = c + s, expanding out p(c + s) as a polynomial in
    s, and then substituting t - c for each occurrence of s.

(a) p(t) = t4 + t2 - 3t + 7 c = 3

(b) p(t) = t5 - 4t3 + 2t2 - 7 c = -5

(c) p(t) = t7 + t6 - t4 + t2 - 5t - 1 c = 6.