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. 👍 0
  2. 👎 0
  3. 👁 92

Respond to this Question

First Name

Your Response

Similar Questions

  1. calculus--please help!!

    I have two questions that I don't understand and need help with. 1. information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zerosof f. degree 4, zeros i;9+i 2. form a polynomial f(x)

    asked by Paul on December 27, 2012
  2. Pre-calculus-check answers

    Write the polynomial equation of least degree that has the roots: -3i, 3i, i, and -i. Answer: (x+3i)(x-3i)(x-i)(x+i)=x^4+x^3-x^3i+9x^2+9x-9xi-9i 2)Determine if the expression 4m^5-6m^8+m+3 is a polynomial in one variable. If so,

    asked by Lucy on August 23, 2008
  3. algebra

    Identify the degree of each term of each polynomial. Then find the degree of the polynomial. xy^3 + 7x^3y^2 - 6xy^4 + 2 I am really having trouble with how this process works. I know the degree of the polynomial is 5 but I don't

    asked by Liamz on January 20, 2015
  4. Math

    Use the given information about a polynomial whose coefficients are real numbers to find the remaining zeros of the polynomial. Degree: 6 Zeros: -6 + 13i^3, -8 + s^2i, -3 - 4i

    asked by CheezyReezy on June 1, 2011
  5. algebra

    Like Terms and Degree Identify the degree of each term of each polynomial. Then find the degree of the polynomial xy^3 + 7x^3y^2 - 6xy^4 + 2

    asked by Liamz on January 16, 2015
  6. Algebra

    Hello, I need some clarification for my answers, thanks! 1. What is the degree of the monomial? A. 5 B. 12*** C. 17 D. 60 2. What is the degree of the monomial? -2 A. -2 B. -3 C. 0*** D. -1 What is the sum or difference? 3. 2x^4 -

    asked by Emma :3 on April 15, 2019
  7. Math

    What does the degree of a polynomial expression tell you about its related polynomial function? Explain your thinking. Give an example of a polynomial expression of degree three. Provide information regarding the graph and zeros

    asked by Seth on January 10, 2018
  8. math

    1.binomial 2.degree of monomial 3.monomial 4.perfect-square trinomial 5.standard form of polynomial A.a polynomial with two terms B.a polynomial in which the terms decrease in degree from left to right and there are no like terms

    asked by ed on May 12, 2010
  9. Algebra

    Please solve as polynomial to third degree: the product of three consecutive odd numbers is 6,783. What are the numbers? The solution being 17, 19, and 21, what are those numbers when considering: x*(x+2)*(x+4)=6,783? Multiplied

    asked by Wayne on March 27, 2013
  10. Algebra

    I already posted this question and Reiny answered it, I just wanted to clarify my answer Problem: If f(x) is a polynomial of degree 4, and g(x) is a polynomial of degree 2, then what is the degree of polynomial f(x)-g(x)? Reiny:

    asked by Trish Goal on November 12, 2014

More Similar Questions