Algebra

Vieta's Formula

Vieta's Formula - Higher Degrees

         

α\alpha, β\beta, and γ\gamma are the roots of the cubic equation x3x2+2x13=0x^3-x^2+2x-13=0. If 1α+1β+1γ=ab\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}=\frac{a}{b}, where aa and bb are coprime positive integers, what is the value of a+ba+b?

Given that 2 roots of f(x)=x3+ax+bf(x) = x^3 + ax + b are 2 and 7, what is a+b a+b ?

What is the product of all roots to the equation

(x1)(x2)(x3)+(x2)(x3)(x4)+(x3)(x4)(x5)+(x4)(x5)(x6)+(x5)(x6)(x7)+(x6)(x7)(x8)=0? \begin{aligned} & (x-1)(x-2)(x-3) + (x-2)(x-3)(x-4) \\ + & (x-3)(x-4)(x-5) + (x-4)(x-5)(x-6) \\ + & (x-5)(x-6)(x-7) + (x-6)(x-7)(x-8) =0 ? \end{aligned}

Details and assumptions

Clarification: Make sure you scroll to the right (if need be) to see the full equation. This problem ends with a "?".

If the quartic x4+3x3+11x2+15x+A x^4 + 3x^3 + 11 x^2 + 15 x + A has roots k,l,m k, l, m and nn such that kl=mnkl = mn , determine AA.

If the cubic equation x33x2+5=0x^3-3x^2+5=0 has three roots α\alpha, β\beta and γ\gamma, what is the value of (α+β)(β+γ)(γ+α)?(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)?

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