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 \({[((6x - 3)x + 9)x + 6]x - 8}x + 12\)
(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).
Answers (1)
Relevant Questions
Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same \(\displaystyle{P}{\left({x}\right)}={3}{x}^{{4}}-{5}{x}^{{3}}+{x}^{{2}}-{3}{x}+{5}\ {Q}{\left({x}\right)}={\left({\left({\left({3}{x}-{5}\right)}{x}+{1}\right)}{x}-{3}\right)}{x}+{5}\) Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial \(\displaystyle{R}{\left({x}\right)}={x}^{{5}}—{2}{x}^{{4}}+{3}{x}^{{3}}—{2}{x}^{{3}}+{3}{x}+{4}\) in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
DISCOVER: Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same \(P(x) = 3x^{4} - 5x^{3} + x^{2} - 3x +5\)
\(Q(x) = (((3x - 5)x + 1)x^3)x + 5\)
Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial
\(R(x) =x^{5} - 2x^{4} + 3x^{3} - 2x^{2} + 3x + 4\) in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head.
Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
Polynomial equation with real coefficients that has the roots \(3, 1 -i\ \text{is}\ x^{3} - 5x^{2} + 8x - 6 = 0.\)
An experiment on the probability is carried out, in which the sample space of the experiment is
\(S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.\)
Let event \(E={2, 3, 4, 5, 6, 7}, event\)
\(F={5, 6, 7, 8, 9}, event\ G={9, 10, 11, 12}, and\ event\ H={2, 3, 4}\).
Assume that each outcome is equally likely. List the outcome s in For G.
Now find P( For G) by counting the number of outcomes in For G.
Determine P (For G ) using the General Addition Rule.
