I am not sure if I titled this correctly. I am working through Calculus With Analytic Geometry by Simmons. Question $\# 17$ on page $87$ reads
Find the equation of the tangent to the curve $y = x^3$ at the point $(a, a^3)$. For what values of a does this tangent intersect the curve at another point?
I can calculate the equation of the tangent line to be $y=3a^2x-2a^3$ but I am not sure how to calculate the values of $a$ where the tangent intersects the curve at another point. Even a hint would be great, thank you.
Best Answer
The values of $x$ where the tangent line and $y=x^3$ intersect are $x^3 = 3a^2x-2a^3$ by substituting $y=x^3$ in.
Now $x^3-3a^2x+2a^3 = 0$. However, you already know that the point $(a,a^3)$ lies on the tangent line and on the curve, so $x-a$ is a factor by the factor theorem.
Then use polynomial long division to get $(x^2+ax-2a^2)(x-a) = 0$, where you can now factor the quadratic.