# 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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