(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.

\(P(2) = {[((6.2 - 3)2 + 9)2 + 6]2 - 8} 2 + 12\)

\(= {[(18 + 9)2+6]2—8}2 + 12\)

\(= {[54 + 6]2 - 8}2 + 12\)

\(= (120 - 8)2 + 12\) 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.