nested multiplication

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A polynomial f(x), such as 2x3−7x2+5x+11, can be evaluated for x=h by calculating h2 and h3, multiplying by appropriate coefficients and summing the terms. But fewer operations are required if the polynomial is rewritten as (2x−7)x+5)x+11 and then evaluated. This method, known as nested multiplication, is therefore more efficient and is recommended when evaluation is to be carried out by hand or by computer. A polynomial a5x5+a4x4+a3x3+a2x2+a1x+a0 of degree 5, for example, would be rewritten for this purpose as(((a5x+a4)x+a3)x+a2)x+a1)x+a0. The steps involved in this evaluation correspond exactly to the calculations that are made in the process of synthetic division, in which the remainder gives f(h).

Subjects: Mathematics.

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