2.1. Horner’s Method 53- Expand the polynomial y7 - 4y6 + 2y4 - y” + y + 1 in terms of (y + 2)
and evaluate it at y = -1.95 to two decimal places. - What does the table for expanding p(t) in powers oft look like?
- Let p, q, r, s be the zeros of the quartic t4 - 3t3 + 2t2 + 5t - 2. Use
Horner’s Method to find a polynomial with integer coefficients whose
zeros arep+3, q+3, r+3, s+3.
Explorations
E.16. We can evaluate 65 by means of four multiplications. However, the
number of multiplications can be reduced to 3:6 x 6 = 36, 36 x 36 = 1296, 6 x 1296 = 7776.In general, for an arbitrary positive integer n and constant c, what is the
minimum number of multiplications necessary to compute c”?
(a) Show that, if n = 2k+1 then c” can be obtained with no more than
L multiplications. Is it possible to get by with fewer multiplications?
(b) Show that c” can be computed using a pocket calculator or a com-
puter by some sequence of the following two operations:(i) multiply the display by c (which can be stored in memory);(ii) square the display.(c) Plan the procedure baaed on (b) which you would use to determine
c51. How many multiplications are required? The binary representation (i.e.
to base 2) of 51 will give a clue as to the order in which operations (i) and
(ii) might be taken.
E.17. For small positive values of the integer n, determine the expansion
of tn in terms of (t - 1) using Horner’s table. Look for patterns, depending
on n and Ic, which govern the coefficients of (t - l)k. Try to find general
formulae for these coefficients. Rewrite the equation you get by making the
substitution t = 1 + x.
E.18. Factorial Powers and Summations. The formulae1+2+...+72= 2 L(n+l) n= 1,2,...l2 + 22 +... + n2 = $n(n + 1)(2n + 1) n= 1,2,...are familiar to many high school students. The task of finding analogous
closed formulae for the sums of higher powers such as cubes and fourth
powers increases in complexity with the exponent. Is there a systematic
way of proceeding in general?