Cubic, Tangent, Circle

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The graph of a cubic function with three real roots is drawn. A circle is then drawn so that the circumference of the circle passes through two adjacent roots. A tangent to the cubic is drawn at the point of the centre of the circle. When drawn accurately, the tangent passes through the first root of the cubic.

Having experienced the result by drawing, the investigation then moves to give a formal proof that the tangent will always pass through the first root of the cubic. The solution requires differentiation, substitution of expressions, and determining the equation of a tangent to a circle.

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