5.1. Approximation of Roots 161
(c) Show that, if (a+ b)/2 is not a zero of p(t), then there is a zero
either between a and (e + b)/2 or else between (a + b)/2 and b.
(d) Let n be a given number. Determine an algorithm which will
produce an interval of length no more than (b - a)2-” which
contains a zero of p(t).- For each of the following polynomials, find consecutive integer values
of the variable at which the values of the polynomial differ in sign.
Between these values, determine by the method of Exercise 1, a zero
to an accuracy of two decimal places
(a) t4 - t3 -P-t-1
(b) 2x3 - 9x2 + 12x + 7
(c) 2t6 - 7t5 + t4 + t3 - 12tz - 5t + 1- Linear interpolation. Refer to the diagram
(a) Find the equation of the straight line joining the points (a,p(a))
and (b,p(b)), and determine its intersection with the z-axis.
(b) Explain why it is reasonable to takep(a)@ - 4
a - p(b) - ~(a)as a first approximation to a zero of p(t).
(c) Devise a modification of the Method of Bisection using (b) and
test it out on the polynomials in Exercise 2.- Homer’s Method. The Horner’s algorithm provides a systematic way
of approximating zeros. Let p(t) = t4 - 3t3 + 7t2 - 15t + 1 and consider
the following chain of Horner’s tables: