Using Horner's rule, evaluate the polynomial p(x)= 2x5 -5x4 -3x2+15 at x=2. Analyse the computation time required for polynomial evaluation using Horner’s rule against the Brute force method.
To evaluate the polynomial at using Horner's rule, we first rewrite the polynomial in a nested form. Horner's rule is an efficient method for evaluating polynomials that minimizes the number of operations, particularly multiplications.
Rewrite the polynomial:
Rewrite using nested multiplication:
Evaluate at :
Calculate each term separately and sum them up:
Calculate each term individually:
Sum them up:
Horner’s Rule:
Brute Force Method:
Horner’s rule reduces the number of multiplications required, as there are no exponentiations required, making it computationally more efficient, especially for polynomials of higher degrees. The efficiency gain is more pronounced when dealing with polynomials with many terms, as it significantly reduces the number of operations needed.
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