Algebra
# Symmetric Polynomials

If \(a,b \ \& \ c\) are the roots of the polynomial \(x^{3}-2x^{2}+3x+1\)

Find the value of \[a^{5}+b^{5}+c^{5}-(a^{4}+b^{4}+c^{4})\]

If \(V_{n}=a^{n}+b^{n},\) where \(a\) and \(b\) are the roots of \(x^{2}+x+1,\) what is the value of

\[ \sum_{n=0}^{1729} (-1)^{n} \cdot \ V_{n} ?\]

\[\Large{P(x) = x^3 - 3x+1}\]

Let \(Q(x) = x^3 + Ax^2 + Bx + C\) be a polynomial with integer coefficients such that its roots are the fifth powers of the roots of \(P(x)\). Evaluate \(A+B+C\).

\[ \large \begin{cases} {a+b+c=9} \\ {a^2+b^2+c^2=99} \\ {a^3+b^3+c^3 = 999} \end{cases} \]

If \(a,b\) and \(c\) are complex numbers that satisfy the system of equations above, find the remainder of \(a^5+b^5+c^5\) when divided by 78.

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