Symmetric Polynomials

Symmetric Polynomials: Level 5 Challenges


If a,b & ca,b \ \& \ c are the roots of the polynomial x32x2+3x+1x^{3}-2x^{2}+3x+1

Find the value of a5+b5+c5(a4+b4+c4)a^{5}+b^{5}+c^{5}-(a^{4}+b^{4}+c^{4})

If Vn=an+bn,V_{n}=a^{n}+b^{n}, where aa and bb are the roots of x2+x+1,x^{2}+x+1, what is the value of

n=01729(1)n Vn? \sum_{n=0}^{1729} (-1)^{n} \cdot \ V_{n} ?

Find the value of a-a for which the roots x1,x2,x3x_{1}, x_{2}, x_{3} of x36x2+axa=0x^{ 3 }-6x^{2}+ax-a = 0 satisfy (x13)3+(x23)3+(x33)3=0\left( x_{1}-3 \right)^{3}+\left(x_{2}-3 \right)^{3}+\left(x_{3}-3 \right)^{ 3 } = 0.

P(x)=x33x+1\Large{P(x) = x^3 - 3x+1}

Let Q(x)=x3+Ax2+Bx+CQ(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)P(x). Evaluate A+B+CA+B+C.

{a+b+c=9a2+b2+c2=99a3+b3+c3=999 \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,ba,b and cc are complex numbers that satisfy the system of equations above, find the remainder of a5+b5+c5a^5+b^5+c^5 when divided by 78.


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