These use divided differences, and are described in most elementary texts on numerical analysis. But please be sure to work with intervals near 0. It is important to evaluate polynomials quickly. For example, we may need to evaluate a polynomial at 100 points in drawing a graph. This can be done very quickly by nested multiplication.
Nested multiplication is just a clever use of brackets, and is best illustrated by example.
Consider the polynomial in the previous example. The main advantage of this neat approach is that we have reduced the number of multiplications. Secondly storage is reduced as we do not have to store the values of the c k 's. Nested multiplication is used in the internal Matlab routine polyval , but this routine has further complications to improve speed. We can prove that this is never the case.