Calculate $x^4 + y^4 + z^4$ if $x + y + z = 1$ , $x^2 + y^2 + z^2 = 2$ and $x^3 + y^3 + z^3 = 3$

Question

I saw this problem here https://www.youtube.com/watch?v=4FNCIYD8HdA.

Can anybody explain why I am getting different results using the following method.

Notations:

S(x⁴) = sum of all terms (each having coefficient 1) consisting of only one variable to the power 4 = x⁴ + y⁴ + z⁴ = A

S(x³y) = sum of all terms (each having coefficient 1) consisting of 2 variables where one of them has power 3 and the other one has power 1 = x³y + x³z + y³x + y³z + …. = B

Similarly, S(x²y²) = C
and S(x²yz) = D

1⁴ = (x + y + z)⁴ = S(x⁴) + 4S(x³y) + 6S(x²y²) + 12S(x²yz) [you can use this website to verify if you have doubt: https://www.mathportal.org/calculators/polynomials-solvers/polynomials-expanding-calculator.php]

Or, 1 = A + 4B + 6C + 12D …………………………………….(1)

2² = (x² + y² + z²)² = S(x⁴) + 2S(x²y²)

Or, 4 = A + 2C ………………………………………………..(2)

3.1 = (x³ + y³ + z³)(x + y + z) = S(x⁴) + S(x³y)

Or, 3 = A + B …………………………………………………...(3)

1².2 = (x + y + z)²(x² + y² + z²) = S(x⁴) + S(x²y²) + 2S(x³y) + 2S(x²yz)

Or, 2 = A + C + 2B + 2D ………………………………………..(4)

Eq(4) - Eq(3) gives

-1 = C + B + 2D ………………………………………………….(5)

Eq(1) - Eq(3) gives

-2 = 3B + 6C + 12D

Or, -⅔ = B + 2C + 4D …………………………………………….(6)

Eq(6) - Eq(5) gives

-⅔ + 1 = C + 2D

Or, ⅓ = C + 2D ……………………………………………………(7)

Eq(1) - Eq(2) gives

-3 = 4B + 4C + 12D

Or, -¾ = B + C + 3D ………………………………………………(8)

Eq(8) - Eq(7) gives

-¾ - ⅓ = B + D

Or, -13/12 = B + D

Or, -13/6 = 2B + 2D …………………………………………….(9)

Eq(4) - Eq(9) gives

2 + 13/6 = A + C or, A + C = 25/6

Or, 25/3 = 2A + 2C ………………………………………………..(10)

Eq(10) - Eq(2) gives

A = 25/3 - 4 = 13/3

Ans is x4 + y4 + z4 = 13/3 . But in this video https://www.youtube.com/watch?v=4FNCIYD8HdA They say the answer is 25/6. I didn’t understand nor want to understand how they reached that solution. All I want to know if my answer 13/3 is correct or wrong. If wrong then please explain why my method gives incorrect result.

I tried to solve by similar method for x5 + y5 + z5 and got different result from what is shown in the video.

Answer 1

You should have written

$1^2\cdot2 = (x + y + z)^2(x^2 + y^2 + z^2 ) = S(x^4) +\color{red}2 S(x^2 y^2 ) + 2S(x^3 y) + 2S(x^2 yz);$

with that, you should get the correct answer.

Answer 2

We have in general if $$ s_1:=x+y+z\textrm{, }s_2:=x^2+y^2+z^2\textrm{, }s_3:=x^3+y^3+z^3 $$ and $$ s_4:=x^4+y^4+z^4 $$ and if $$ \sigma_1=x+y+z\textrm{, }\sigma_2=xy+yz+zx\textrm{, }\sigma_3=xyz\tag 1 $$ Then $$ s_1=\sigma_1\textrm{, }s_2=\sigma_1^2-2\sigma_2\textrm{, }s_3=\sigma_1^3-3\sigma_1\sigma_2+3\sigma_3\textrm{, }s_4=\sigma_1^4-4\sigma_1^2\sigma_2+2\sigma_2^2+4\sigma_1 \sigma_3\tag 2 $$ Hence $$ \sigma_1=s_1=1\textrm{, }\sigma_2=-1/2\textrm{, }\sigma_3=1/6 $$ Hence $$ s_4=x^4+y^4+z^4=25/6 $$