MathDB

47

Part of 1970 IMO Longlists

Problems(1)

The ugly polynomial divisibility

Source: IMO LongList 1970 - P47

5/22/2011
Given a polynomial P(x)=ab(a−c)x3+(a3−a2c+2ab2−b2c+abc)x2+(2a2b+b2c+a2c+b3−abc)x+ab(b+c),P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c), where a,b,c≠0a, b, c \neq 0, prove that P(x)P(x) is divisible by Q(x)=abx2+(a2+b2)x+abQ(x) = abx^2 + (a^2 + b^2)x + ab and conclude that P(x0)P(x_0) is divisible by (a+b)3(a + b)^3 for x0=(a+b+1)n,n∈Nx_0 = (a + b + 1)^n, n \in \mathbb N.
algebrapolynomialcalculusderivativealgebra unsolved