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Symmetric Polynomials

In a symmetric polynomial, you can interchange any of the variables and get the same polynomial. Symmetric polynomials form the basis of Galois theory, which connects group theory and field theory.

Challenge Quizzes

Level 5

         

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

Find the value of \(-a\) for which the roots \(x_{1}, x_{2}, x_{3}\) of \(x^{ 3 }-6x^{2}+ax-a = 0\) satisfy \(\left( x_{1}-3 \right)^{3}+\left(x_{2}-3 \right)^{3}+\left(x_{3}-3 \right)^{ 3 } = 0\).

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

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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