Bolzano's Theorem + Rolle's Theorem

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Show that the equation

2x³ + 3x² + 6x + 1 = 0,

has exactly one real root.

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Answer

Let f (x) = 2x³ + 3x² + 6x + 1. We have f (0) = 1  and  f(– 1) = – 4. So the Intermediate Value Theorem (Bolzano's Theorem) shows that there exists a point ξÎ(– 1,0) such that f (ξ) = 0.

Consequently our equation has at least one real root.

Let us now show that this equation has also at most one real root. Assume not, then there must exist at least two roots ρ₁and ρ₂, with ρ₁ < ρ₂.
Then we have f (ρ₁) = 0  and  f (ρ₂) = 0. Rolle's Theorem implies the existence of a point x₀Î(ρ₁,ρ₂) such that

f '(x₀) = 6x₀² + 6x₀ + 6 =0.

But the quadratic equation 6x₀² + 6x₀ + 6 = 0 does not have real roots, yielding a contradiction to our assumption that f (x) had at least two roots. Conclusion: our original equation has exactly one real root.