Isosceles Cubic Teaching Tool

Algebra Level pending

Consider the general integer cubic polynomial \(f(x)=ax^3+bx^2+cx+d\) and let \(a>1\), \(c=-a.\) The vertices \((0,d)\), \((1,f(1))\), \((-1,f(-1))\) form an isosceles triangle since \(f(1)=f(-1)=b+d\). Let \(d>0\) so this triangle is standing up (Legs). Then \(f(1)<0\) implies \(f(x)\) has a root in the interval \((-1,1)\). Show that this root is rational, and find the area function of this isosceles triangle \(A(a,d)\).

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