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 than evaluating a polynomial with powers. Formula used: Expansion of a polynomial, Suppose, an equation is given in the form \(a(x + 2) x\) can be expanded and written as, \(a(x+2)x = ax^2 + 2ax.\) This is known as expansion of an equation. Calculation: The given equation \({[(3x + 5)x + 4]x + 3} x + 1\) can be expanded as follow, \({[(3x +5)x + 4]x + 3} x + 1 = {[3x^2 + 5x + 4]x + 3} x + 1\)

\(={3x^3 + 5x^2 + 4x + 3}x + 1\)

\(=3x^4 + 5x^3 + 4x^2 + 3x + 1\) Therefore, the equation \({[(3x + 5)x + 4]x + 3} x + 1 = 3x^4 + 5x^3 + 4x^2 + 3x + 1\) isan identifity. (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. Formula used: Grouping of similar terms in a polynomial, Suppose, an equation is given in the form \(ax^2 + 2ax + 3\) can be grouped and written as, \(ax^2 + 2ax +3 = ax (x + 2) + 3 =a(x + 2)x + 3.\) This is known as grouping of an equation. Calculation: The polynomial, \(P(x) = 6x^5 — 3x^4 + 9x^3 + 6x^2 — 8x + 12,\) is grouped by the similar terms and written as follow,

\(P(x) = 6x^5 — 3x^4 + 9x^3 + 6x^2 — 8x + 12 = x(6x^4 - 3x^3 + 9x^2 -8x) + 12\)

\(= x (x(6x^3 - 3x^2 + 9x + 6)- 8)+ 12\)

\(={x[x(x(6x^2 - 3x + 9)+ 6)- 8]}+ 12\) Furhter, \({x[x(x(6x^2 - 3x + 9)+ 6)- 8]}+ 12 = {[((6x - 3)x + 9)x + 6]x - 8}x + 12\) Therefore, the polynomial \(P(x) = 6x^5 — 3x^4 + 9x^3 + 6x^2 — 8x + 12\) without powers is \({[((6x - 3)x + 9)x + 6]x - 8}x + 12\)