(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.

(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,
$P(2)=[((6.2-3)2+9)2+6]2-82+12$
$=[(18+9)2+6]2—82+12$
$=[54+6]2-82+12$
$=(120-8)2+12$ 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.