The Intermediate Value Theorem from first-year calculus says that if is continuous on the closed interval and are nonzero and opposite signs, then has a root in the open interval . We have and , so there is indeed a root in . The graph in Figure 16.6.2 was obtained by performing a binary search by splitting into subintervals.
Figure16.6.2.A binary search tree search for the root of a polynomial.
Since , the root must be in the subinterval . This tells us to round down to instead of rounding up to , so we conclude that the root is approximately .
