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
sol=Solve[...]
and then1. (a + b + c) /. sol
. $\endgroup$