If a + b + c = 0 and a 2 + b 2 + c 2 = 1, work out a 4 + b 4 + c 4 .Could this problem admit a solution through the method of lagrange multipliers.

Question

If   a  +  b  +  c  =  0 and       a    2    +      b    2    +      c    2    =  1, work out       a    4    +      b    4    +      c    4  .Could this problem admit a solution through the method of lagrange multipliers.

Zara Pratt · Accepted Answer

Let   a  ,  b  ,  c be the roots of a cubic polynomial.Since   a  +  b  +  c  =  0 this has form       x    3    +  p  x  +  q  =  0 where  p  =  a  b  +  b  c  +  c  a  =                                                  (          a          +          b          +          c                      )            2                    −          (                      a            2                    +                      b            2                    +                      c            2                    )                                                                  2                      =  −                                                  1                                                                  2                    Now note multiply by   x to see that   a  ,  b  ,  c satisfy   f  (  x  )  =      x    4    +  p      x    2    +  q  x  =  0Then  0  =  f  (  a  )  +  f  (  b  )  +  f  (  c  )  =  (      a    4    +      b    4    +      c    4    )  +  p  (      a    2    +      b    2    +      c    2    )  +  q  (  a  +  b  +  c  )Substituting known values gives       a    4    +      b    4    +      c    4    =                                                  1                                                                  2                    Working with polynomials is sometimes a convenient way of capturing and organising information about symmetric functions.

If a+b+c=0 and a^2+b^2+c^2=1, work out a^4+b^4+c^4. Could this problem admit a solution through the method of lagrange multipliers

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