Am I missing anything? Solving Equations

I want to know what $a^4+b^4+c^4$ would be when

$a+b+c=1, a^2+b^2+c^2=2, a^3+b^3+c^3=3.$

I tried

Solve[{a + b + c == 1, a^2 + b^2 + c^2 == 2, 
a^3 + b^3 + c^3 == 3}, {a, b, c}]


a^4 + b^4 + c^4 /. {a -> ... }

I copied and pasted one instance from the result of Solve[] into {a -> ...}

The solution and the result of Simplify[] of it was not quite what I expected.

So as a test, I tried a+b+c /. {a -> ...}

Still it was strange, it didn't come up with 1 which is obvious correct answer.

So am I missing anything here?

Is there any better way?

Answer

Solve your equations with respect to another variable d == a^4 + b^4 + c^4 eliminating given ones

Solve[{ a + b + c == 1,       a^2 + b^2 + c^2 == 2, 
        a^3 + b^3 + c^3 == 3, a^4 + b^4 + c^4 == d},  {d}, {a, b, c}]

{{d -> 25/6}}

This tutorial will be helpful Eliminating Variables.

Edit

For the sake of completeness we add another methods of dealing with the problem. Let's point out even simpler way than the first one, just add assumptions to Simplify:

Simplify[ a^4 + b^4 + c^4, {a + b + c == 1, a^2 + b^2 + c^2 == 2, a^3 + b^3 + c^3 == 3}]

25/6

Of course we can use FullSimplify as well. In this case, Simplify works nicely, but the first method is stronger and should work in more general cases.

SymmetricReduction is more appropriate when we would like to get symbolic results, i.e. here we would like to express a^4 + b^4 + c^4 in terms of these polynomials: {a + b + c, a^2 + b^2 + c^2, a^3 + b^3 + c^3}

First we need to get rid of unknown symmetric polynomials (we know only a + b + c):

 SymmetricPolynomial[#, {a, b, c}] & /@ Range[3]

{a + b + c, a b + a c + b c, a b c}

form the last reduction using the previous ones:

First @ SymmetricReduction[ a^# + b^# + c^#, {a, b, c}] & /@ Range[4] // Column

a + b + c

(a + b + c)^2 - 2 (a b + a c + b c)


3 a b c + (a + b + c)^3 - 3 (a + b + c) (a b + a c + b c) 

4 a b c (a + b + c) + (a + b + c)^4 - 4 (a + b + c)^2 (a b + a c + b c)
+ 2 (a b + a c + b c)^2

Now we have a^4 + b^4 + c^4 expressed in terms of given polynomials:

str = 
First[ SymmetricReduction[ a^4 + b^4 + c^4, {a, b, c}]] //. {
  (a b + a c + b c) -> 1/2 (a + b + c)^2 - 1/2 (a^2 + b^2 + c^2),
  (a b c) -> 1/3 (a^3 + b^3 + c^3) - 1/3 (a + b + c)^3 + (a + b + c) (a b + a c + b c)}

 (a + b + c)^4 - 4 (a + b + c)^2 (1/2 (a + b + c)^2 + 1/2 (-a^2 - b^2 - c^2))
 + 2 (1/2(a + b + c)^2 + 1/2(-a^2 - b^2 - c^2))^2 + 4(a + b + c) (-(1/3)(a + b + c)^3
 + 1/3 (a^3 + b^3 + c^3) + (a + b + c) (1/2 (a + b + c)^2 + 1/2 (-a^2 - b^2 - c^2)))

of course

Simplify @ %

a^4 + b^4 + c^4

and finally:

str /. {a + b + c -> 1, (-a^2 - b^2 - c^2) -> -2, (a^3 + b^3 + c^3) -> 3}

25/6