Question

Start with g add x then divide by a this is sons homework needs help ASAP please

Answer 1

First note that

x2+y2+z2=(x+y+z)2−2(xy+yz+zx)=35

Now substituting x+y+z=1, we have 12−2(xy+yz+zx)=35, and thus xy+yz+zx=−17.

Then,

x3+y3+z3=(x+y+z)3−3(x+y+z)(xy+yz+zx)+3xyz=97

and substituting with 1 and our above result once again leads us to have xyz=15. Now we need a product of 15 and sum of 1 for (x,y,z), giving us (−1,−3,5) as a solution, amongst others.

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Jul 13 '15 at 15:46

miradulo

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Jul 13 '15 at 15:49

1

This is a very nice answer! +1 – Hassan Muhammad Jul 13 '15 at 19:42

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That problem has a unique solution, up to permutations of the variables.

That happens since the power sums pk=xk+yk+zk for k=1,2,3 give you the values of the elementary symmetric functions e1=x+y+z,e2=xy+xz+yz,e3=xyz through Newton's identities. Then x,y,z can be identified with the roots of the polynomial:

p(w)=w3−e1w2+e2w−e3.

If you know in advance that (x,y,z)=(−1,−3,5) works, then every solution is given by a permutation of {−1,−3,5}, since the coefficients of p(w) are always the same, as well as its roots:

p(w)=w3−w2−17w−15=(w+1)(w+3)(w−5).