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Problem 2. The following algorithm which is used to evaluate a polynomial anx" + an-1x" -1 ++ a₁x + ªo at x = c is expressed in pseudocode as follows: procedure polynomial evaluation (c, ao,, an: real numbers) y := an for i:=1 to n y := y*c+an-i return y {y = anx² + an−1x¹−¹ + · + a₁x + ao}. where the final value of y is the value of the polynomial at x = c. (a) Evaluate 12x4 + 3x³ − 5x² + 7x - 8 at x = -1 by working through each step of the algorithm showing the values assigned at each assignment step CLEARLY. Answers with missing iterations, indices, etc. will not be taken into account. (b) How many additions and multiplications take place in the evaluation of a polynomial of degree n? Briefly explain. (Do not include the additions needed to increment the loop variable i.) (c) One can represent polynomials using nested multiplications, that is successively taking à as a common factor in the remaining polynomials of decreasing degrees, for example 3x³ + 2x² + 5x + 4 = x(x(3x + 2) + 5) +4. Represent the given polynomial given in (a) using nested multiplications and briefly explain the con- nection between the presented algorithm and this representation.

Problem 2. The following algorithm which is used to evaluate a polynomial anx" + an-1x" -1 ++ a₁x + ªo at x = c is expressed in pseudocode as follows: procedure polynomial evaluation (c, ao,, an: real numbers) y := an for i:=1 to n y := y*c+an-i return y {y = anx² + an−1x¹−¹ + · + a₁x + ao}. where the final value of y is the value of the polynomial at x = c. (a) Evaluate 12x4 + 3x³ − 5x² + 7x - 8 at x = -1 by working through each step of the algorithm showing the values assigned at each assignment step CLEARLY. Answers with missing iterations, indices, etc. will not be taken into account. (b) How many additions and multiplications take place in the evaluation of a polynomial of degree n? Briefly explain. (Do not include the additions needed to increment the loop variable i.) (c) One can represent polynomials using nested multiplications, that is successively taking à as a common factor in the remaining polynomials of decreasing degrees, for example 3x³ + 2x² + 5x + 4 = x(x(3x + 2) + 5) +4. Represent the given polynomial given in (a) using nested multiplications and briefly explain the con- nection between the presented algorithm and this representation.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Problem 2. The following algorithm which is used to evaluate a polynomial Anx" + an-1x"–1 +...+ ajx+ ao at x = c is expressed in pseudocode as follows: procedure polynomial evaluation (c, ao, · , an : real numbers) y := an for i := 1 to n y := y * c + ɑn-i return y {y = anx" + an-1x"-1 +...+ a1x + ao}. where the final value of y is the value of the polynomial at x = c. (a) Evaluate 12x4 + 3x³ – 5a² + 7x – 8 at x = -1 by working through each step of the algorithm showing the values assigned at each assignment step CLEARLY. Answers with missing iterations, indices, etc. will not be taken into account. (b) How many additions and multiplications take place in the evaluation of a polynomial of degree n? Briefly explain. (Do not include the additions needed to increment the loop variable i.) (c) One can represent polynomials using nested multiplications, that is successively taking x as a common factor in the remaining polynomials of decreasing degrees, for example 3x3 + 2x? + 5x +4= x(x(3x + 2) + 5) +4. Represent the given polynomial given in (a) using nested multiplications and briefly explain the con- nection between the presented algorithm and this representation.
Transcribed Image Text:Problem 2. The following algorithm which is used to evaluate a polynomial Anx" + an-1x"–1 +...+ ajx+ ao at x = c is expressed in pseudocode as follows: procedure polynomial evaluation (c, ao, · , an : real numbers) y := an for i := 1 to n y := y * c + ɑn-i return y {y = anx" + an-1x"-1 +...+ a1x + ao}. where the final value of y is the value of the polynomial at x = c. (a) Evaluate 12x4 + 3x³ – 5a² + 7x – 8 at x = -1 by working through each step of the algorithm showing the values assigned at each assignment step CLEARLY. Answers with missing iterations, indices, etc. will not be taken into account. (b) How many additions and multiplications take place in the evaluation of a polynomial of degree n? Briefly explain. (Do not include the additions needed to increment the loop variable i.) (c) One can represent polynomials using nested multiplications, that is successively taking x as a common factor in the remaining polynomials of decreasing degrees, for example 3x3 + 2x? + 5x +4= x(x(3x + 2) + 5) +4. Represent the given polynomial given in (a) using nested multiplications and briefly explain the con- nection between the presented algorithm and this representation.
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