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Question # (a)To calculate: The following equation {[(3x + 5)x + 4]x + 3} x + 1 = 3x^4 + 5x^3 + 4x^2 + 3x + 1 is an identity,(b) To calculate: The lopynomial P(x) = 6x^5 - 3x^4 + 9x^3 + 6x^2 -8x + 12 without powers of x as in patr (a).

Polynomial arithmetic
ANSWERED (a)To calculate: The following equation $${[(3x + 5)x + 4]x + 3} x + 1 = 3x^4 + 5x^3 + 4x^2 + 3x + 1$$ is an identity, (b) To calculate: The lopynomial $$P(x) = 6x^5 - 3x^4 + 9x^3 + 6x^2 -8x + 12$$ without powers of x as in patr (a). 2020-10-29

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 $${[((6x - 3)x + 9)x + 6]x - 8}x + 12$$