Roots α, β of 2x²+cosθ·x−1=0 depend on θ∈(0,2π). Find 16(M+m) where m, M are the minimum and maximum values of α⁴+β⁴ over all θ.
Two positive numbers p, q satisfy p+q=2 and p^4+q^4=272. Find pq using power-sum identities, then write the quadratic having p, q as roots.
Find α³+β³+γ³+δ³ for the roots of x⁴−3x³−2x²+3x+1=0 by factoring the quartic into two quadratics, avoiding solving for the roots directly.
Find the product of all rational roots of (x²−9x+11)²−(x−4)(x−5)=3 by substituting y=x²−9x+11 to reduce it to a simple quadratic in y.
Find the positive square root of 3+√5 by expressing it as √x+√y, splitting the equation into a rational and irrational part, and solving for x and y.
Let α, β (α>β) be roots of x²−√2x−√3=0 and Pn=αⁿ−βⁿ. Find (11√3−10√2)P10+(11√2+10)P11−11P12 using the Newton recurrence for power sums.