(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).

Dolly Robinson 2020-10-28 Answered

(a)To calculate: The following equation [(3x+5)x+4]x+3x+1=3x4+5x3+4x2+3x+1 is an identity, (b) To calculate: The lopynomial P(x)=6x53x4+9x3+6x28x+12 without powers of x as in patr (a).

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Expert Answer

mhalmantus
Answered 2020-10-29 Author has 106 answers

Given: A fourth-degree polynomial in x such as 3x4+5x3+4x2+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=ax2+2ax. This is known as expansion of an equation. Calculation: The given equation [(3x+5)x+4]x+3x+1 can be expanded as follow, [(3x+5)x+4]x+3x+1=[3x2+5x+4]x+3x+1
=3x3+5x2+4x+3x+1
=3x4+5x3+4x2+3x+1 Therefore, the equation [(3x+5)x+4]x+3x+1=3x4+5x3+4x2+3x+1 isan identifity. (b) Given: A fourth-degree polynomial in x such as 3x4+5x3+4x2+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 ax2+2ax+3 can be grouped and written as, ax2+2ax+3=ax(x+2)+3=a(x+2)x+3. This is known as grouping of an equation. Calculation: The polynomial, P(x)=6x53x4+9x3+6x28x+12, is grouped by the similar terms and written as follow,
P(x)=6x53x4+9x3+6x28x+12=x(6x43x3+9x28x)+12
=x(x(6x33x2+9x+6)8)+12
=x[x(x(6x23x+9)+6)8]+12 Furhter, x[x(x(6x23x+9)+6)8]+12=[((6x3)x+9)x+6]x8x+12 Therefore, the polynomial P(x)=6x53x4+9x3+6x28x+12 without powers is [((6x3)x+9)x+6]x8x+12

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