Posted by **Count Iblis** on Monday, July 23, 2007 at 2:42am.

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?

## Answer this Question

## Related Questions

- math - Logarithm!!! Select all of the following that are true statements: (a) ...
- math - write as a single logarithm: -2logbase3(1/x)+(1/3)logbase3(square root of...
- Mathematics - Prove that log a, log ar, log ar^2 is an a.p. Is the following ...
- college algebra - Write expression as one logarithm and simplify if appropriate...
- maths - which three statements are true? a) if x= -10^4 then log 10 = -4 b)if x...
- maths - which three statements are true? a) if x= -10^4 then log 10 = -4 b)if x...
- Advanced Functions/ Precalculus Log - 1. Evaluate 4^(log base4 64) + 10^(log100...
- Math - Logarithm - Solve the following logarithm equation. Please show all of ...
- Math Help Please - Which of the following expressions is equal to log (x sqrt-y...
- math - write as a single logarithm: ln(x/x-1)+ ln(x+1/x)- ln(x^2-1) please show ...

More Related Questions