quality:correction, information:extraction, linguistic:translation
Example: Finding the root of a polynomial

Suppose that the bisection method is used to find a root of the polynomial

First, two numbers a {\displaystyle a} and b {\displaystyle b} have to be found such that f (a) {\displaystyle f(a)} and f (b) {\displaystyle f(b)} have opposite signs. For the above function, a = 1 {\displaystyle a=1} and b = 2 {\displaystyle b=2} satisfy this criterion, as

and

Because the function is continuous, there must be a root within the interval.

In the first iteration, the end points of the interval which brackets the root are a 1 = 1 {\displaystyle a_{1}=1} and b 1 = 2 {\displaystyle b_{1}=2}, so the midpoint is

The function value at the midpoint is f (c 1) = (1.5) 3 − (1.5) − 2 = − 0.125 {\displaystyle f(c_{1})=(1.5)^{3}-(1.5)-2=-0.125}. Because f (c 1) {\displaystyle f(c_{1})} is negative, a = 1 {\displaystyle a=1} is replaced with a = 1.5 {\displaystyle a=1.5} for the next iteration to ensure that f (a) {\displaystyle f(a)} and f (b) {\displaystyle f(b)} have opposite signs. As this continues, the interval between a {\displaystyle a} and b {\displaystyle b} will become increasingly smaller, converging on the root of the function. See this happen in the table below.

After 13 iterations, it becomes apparent that there is a convergence to about 1.521: a root for the polynomial.