(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).

Question

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

Answer 1

Given: A fourth-degree polynomial in x such as $3 x^{4} + 5 x^{3} + 4 x^{2} + 3 x + 1$ 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, $a (x + 2) x = a x^{2} + 2 a x .$ This is known as expansion of an equation. Calculation: The given equation $[(3 x + 5) x + 4] x + 3 x + 1$ can be expanded as follow, $[(3 x + 5) x + 4] x + 3 x + 1 = [3 x^{2} + 5 x + 4] x + 3 x + 1$
$= 3 x^{3} + 5 x^{2} + 4 x + 3 x + 1$
$= 3 x^{4} + 5 x^{3} + 4 x^{2} + 3 x + 1$ Therefore, the equation $[(3 x + 5) x + 4] x + 3 x + 1 = 3 x^{4} + 5 x^{3} + 4 x^{2} + 3 x + 1$ isan identifity. (b) Given: A fourth-degree polynomial in x such as $3 x^{4} + 5 x^{3} + 4 x^{2} + 3 x + 1$ 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 $a x^{2} + 2 a x + 3$ can be grouped and written as, $a x^{2} + 2 a x + 3 = a x (x + 2) + 3 = a (x + 2) x + 3.$ This is known as grouping of an equation. Calculation: The polynomial, $P (x) = 6 x^{5} — 3 x^{4} + 9 x^{3} + 6 x^{2} — 8 x + 12,$ is grouped by the similar terms and written as follow,
$P (x) = 6 x^{5} — 3 x^{4} + 9 x^{3} + 6 x^{2} — 8 x + 12 = x (6 x^{4} - 3 x^{3} + 9 x^{2} - 8 x) + 12$
$= x (x (6 x^{3} - 3 x^{2} + 9 x + 6) - 8) + 12$
$= x [x (x (6 x^{2} - 3 x + 9) + 6) - 8] + 12$ Furhter, $x [x (x (6 x^{2} - 3 x + 9) + 6) - 8] + 12 = [((6 x - 3) x + 9) x + 6] x - 8 x + 12$ Therefore, the polynomial $P (x) = 6 x^{5} — 3 x^{4} + 9 x^{3} + 6 x^{2} — 8 x + 12$ without powers is $[((6 x - 3) x + 9) x + 6] x - 8 x + 12$

(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).

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