(a)To calculate: The value of P(2) without a calculator using both forms of the polynomial. The value of P (2) is 236. (b)The least amount of arithmetic operations was performed in (c) using Horner's method.

Question
Polynomial arithmetic
asked 2021-03-18
(a)To calculate: The value of P(2) without a calculator using both forms of the polynomial. The value of P (2) is 236. (b)The least amount of arithmetic operations was performed in (c) using Horner's method.

Answers (1)

2021-03-19
(a) Given: A fourth-degree polynomial in x such as 3x^4 + 5x^3 + 4x^2 + 3x + 1 contains all of the powers of x from the first through the fourth. However, any polynomial can be written without powers of x. Evaluating a polynomial without powers of x (Horner's method) is somewhat easier thanevaluating a polynomial with powers. Calculation: The polynomial, \(P(x) = 6x^5 — 3x^4 + 9x^3 + 6x^2 — 8x + 12,\) can be written as, \({[((6x — 3)x + 9)x + 6]x — 8) x + 12\) without its power (Horner's method), and value of P(2) is given by,
\(P(2) = {[((6.2 - 3)2 + 9)2 + 6]2 - 8} 2 + 12\)
\(= {[(18 + 9)2+6]2—8}2 + 12\)
\(= {[54 + 6]2 - 8}2 + 12\)
\(= (120 - 8)2 + 12\) Further, \(P(2) = 6(2)^5 - 3(2)^4 + 9(2)^3 + 6(2)^2 - 8(2) + 12\)
\(= 6.32 - 3.16 + 9.8 + 6.4 - 16 + 12\)
\(= 192 - 48 + 72 + 24 - 4\)
\(=236\) Therefore, the value of P(2)is 236. (b) Given: A fourth-degree polynomial in x such as \(3x^4 + 5x^3 + 4x^2 + 3x + 1\) contains all of the powers of xfrom the first through the fourth. However, any polynomial can be written without powers of x. Evaluating a polynomial without powers of x (Horner's method) is somewhat easier than evaluating a polynomial with powers. Calculation: Horner's method is the method of writing a polynomial without the powers of x and in its simplest form which makes the calculation much easier. therefore, the least amount of airhmetic operations was performed in (c) using Horner's method.
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