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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.

# 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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