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struggling to see how to express the other terms in terms of a, b and c

Looks good so far.

isn't the answer just
P(x)=(x-r1^4)(x-r2^4)(x-r3^4)

That's right.

The coefficients of the polynomial we're looking for are to be expressed in terms of a, b, and c.

tough one isn't it, jimmy?

too right

best of luck! - must go - i have to take the wife shopping.

no worries, have fun

lol - u must be joking! - i'm spending money!

i give up. do you have a solution abtrehearn?

\[x^3-x^2*(a^4+R_a)+x(b^4-R_b)-c^4\]

lol
\[R_a\] means the remaining part of a^4
\[R_b\] means the remaining part of b^4

the remaining part?

so there should be a negative in front of R_a

since there is already a negative in front of that parenthesis

yes, but the gibberish is in terms of r1, r2 and r3

(r1+r2+r3)^4=r1^4+r2^4+r3^4+R(x)

This identity leads to
\[a ^{2} = r _{1}^{2} + r _{2}^{2} + r _{3}^{2} + 2b.\]

So a^2 - 2b can sub in for r1^2 + r2^2 + r3^2 in the cubic identity.