# (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)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) usingHorner's method.
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(a) Given: A fourth-degree polynomial in x such as $3{x}^{4}+5{x}^{3}+4{x}^{2}+3x+1$ 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, $P\left(x\right)=6{x}^{5}—3{x}^{4}+9{x}^{3}+6{x}^{2}—8x+12,$ can be written as, $\left[\left(\left(6x-3\right)x+9\right)x+6\right]x-8\right)x+12$ without its power (Horner's method), and value of P(2) is given by,
$P\left(2\right)=\left[\left(\left(6.2-3\right)2+9\right)2+6\right]2-82+12$
$=\left[\left(18+9\right)2+6\right]2—82+12$
$=\left[54+6\right]2-82+12$
$=\left(120-8\right)2+12$ Further, $P\left(2\right)=6\left(2{\right)}^{5}-3\left(2{\right)}^{4}+9\left(2{\right)}^{3}+6\left(2{\right)}^{2}-8\left(2\right)+12$
$=6.32-3.16+9.8+6.4-16+12$
$=192-48+72+24-4$
$=236$ Therefore, the value of P(2)is 236. (b) Given: A fourth-degree polynomial in x such as $3{x}^{4}+5{x}^{3}+4{x}^{2}+3x+1$ 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.