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The cubic curve $C$ has equation

y = a x^{3} + b x^{2} + c x + d, a \neq 0

The tangent to the cubic at the point where $x = k$ is $ℓ$ .

Prove that, if $x = k$ is the point of inflection of the cubic, then $ℓ$ and $C$ intersect only at the point of tangency.

It is not enough to draw a sketch. We need the equation of $ℓ$ .

To find $ℓ$ , use

y - y_{0} = m (x - x_{0})

with

\begin{aligned} x_{0} & = k \\ y_{0} & = f (k) \\ m & = f^{'} (k) \end{aligned}

Ugly as it may be, the equation of $ℓ$ is

y = (3 a k^{2} + 2 b k + c) x + d - b k^{2} - 2 a k^{3}

The intersections with $C$ occur when

a x^{3} + b x^{2} + c x + d = (3 a k^{2} + 2 b k + c) x + d - b k^{2} - 2 a k^{3}

Since $x = k$ is a root of the equation in the previous hint, we know $(x - k)$ is a factor. How are your long division skills?