maths

Let's denote the three numbers by a1, a2, and a3. Consider the third degree polynomial:

p(x) = (1 + a1 x)(1 + a2 x)(1 + a3 x)

Take the logarithm:

Log[p(x)] =

Log(1 + a1 x) + Log(1 + a2 x)
+ Log(1 + a3 x)

Expand in powers of x by using that:

Log(1 + x) = x - x^2/2 +x^3/3 - x^4/4 + ...

Then, if we denote the sum of the n-th powers of the ai by Sn, we find:

Log[p(x)] = S1 - S2/2 x^2 + S3/3x^3 - S4/4 x^4 + ...

If we now exponentiate this we should get p(x) back:

Exp{Log[p(x)]} = p(x)

We can calculate
Exp[S1 - S2/2 x^2 + S3/3x^3 - S4/4 x^4 + ...]

using the series expansion:

Exp[X] = 1 + X + X^2/2! + X^3/3! + X^4/4! + ...

If we put in here X = S1 - S2/2 x^2 + S3/3x^3 - S4/4 x^4 + ...]

the result must be p(x). Snce p(x) is a third degree polynomial, this means that
all powers of x^4 and higher must vanish.

If you work out the coefficient of x^4 and equate it to zero you get the equation:

S4 = S1^4/6 - S1^2 S2 + S2^2/2 + 4/3 S1 S3

So, if S1 = 4, S2 = 10, and S3 = 22, then S4 must be 50.

the sum of three numbers is 4, the sum of their squares is 10 and the sum of their cubes is 22. what is the sum of their fourth powers?

  1. 👍
  2. 👎
  3. 👁

Respond to this Question

First Name

Your Response

Similar Questions

  1. algebra

    form a polynomial f(x) with real coefficients having the given degree and zeros. Degree 4; zeros:-3 +5i; 2 multiplicity 2 enter the polynomial f(x)=a(?)

  2. Math

    A pair of fair dice is rolled. Let E denote the event that the number landing uppermost on the first die is a 3, and let F denote the event that the sum of the numbers landing uppermost is 6. Determine whether E and F are

  3. Math

    Which of the following statements about a polynomial function is false? 1) A polynomial function of degree n has at most n turning points. 2) A polynomial function of degree n may have up to n distinct zeros. 3) A polynomial

  4. Maths

    true/false 1. a cubic polynomial has at least one zero.............. 2. a quadratic polynomial an have at most two zeroes.......... 3. if r(x)is the remainder and p(x) is the divisor, then degree r(x) < degree p(x)............ 4.

  1. PreCalculus

    Suppose that a polynomial function of degree 5 with rational coefficients has the given numbers as zeros. Find the other zero( s): -1, radical 3, 11/3

  2. algebra

    Write the polynomial in standard form. Then name the polynomial based on its degree and number of terms. 2 – 11x^2 – 8x + 6x^2 A. –5x^2 – 8x + 2; quadratic trinomial B. –5x^2 – 8x; quadratic binomial C. –6x^2 – 8x

  3. Algebra

    Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. 3-i,sqrt2 f(x)=??

  4. Maths

    If a fifth degree polynomial is divide by a third degree polynomial,what is the degree of the quotient

  1. PreCalc

    Find a polynomial of the specified degree that satisfies the given conditions. Degree 4; zeros −3, 0, 1, 4; coefficient of x3 is 4

  2. pre-calc

    1. which of the following is a fourth degree polynomial function? select all that apply. a. f(x)= 4x^3 - x^2 + 2x - 7 b. f(x)= 5-x^4 c. f(x)= 1 / 2x^4 + x^2 -5 d. f(x)= 3x^4 + 2x^3 -4x +1 2. which function below has the end

  3. math

    Find a polynomial f(x) with leading coefficient 1 and having the given degree and zeros. Each polynomial should be expanded from factored form, simplified and written in descending order of exponents on the variable. For example:

  4. Algebra

    Give the degree and classify the polynomial by the number of terms. A. degree: 2; trinomial B. degree: 3; binomial C. degree: 3; trinomial D. degree: 2; binomial Is B correct?

You can view more similar questions or ask a new question.