# Interpretation of the Derivative

a) Show that if $$P(x)$$ is a polynomial such that $$P(a)=P'(a)=0$$ then there exists a polynomial $$Q(x)$$ such that $$P(x)=(x-a)^2Q(x)$$.

b) Show that if $$P(x)$$ is a quartic polynomial then there exists at most one line $$\ell$$ that is tangent to the graph of $$P(x)$$ at two places.

How am I suppose to begin this question? I think I should work backwards from $$P(x)=(x-a)^2Q(x),$$ but I am not sure how to use that $$P(x)$$ and $$Q(x)$$ are polynomials and $$P(a)=P'(a)=0$$ to do this. I think somehow I need to introduce at $$P(a)$$ or $$P'(a)$$ term into that equation, but I am not sure how to do this. Is this correct and the best approach? Please help me figure out what to do next/ how to start.

• What is a polynomial? What is the degree of P? (I know that you don't know exactly what it is, but you know what it isn't) Commented Jan 7, 2020 at 18:29

## 2 Answers

Two hints:

1 - Consider changing the variable from $$x$$ to $$u=x-a$$. So $$P(x)=T(u)$$. Then you'll see if $$T(0)=0$$, then it has no independent term. If $$T'(0)=0$$, then $$T(u)$$ also does not have a first order term. Can you continue afterwards?

2 - Consider a quartic of the form $$P(x) = a x^4+b x^3 + c x^2 + dx +e$$, then consider a line of the form $$L(x) = kx+m$$. Try solving for $$L(x)=P(x)$$ and $$L'(x)=P'(x)$$ simultaneously.

Here's another approach for the first question — using the factor theorem (a special case of the polynomial remainder theorem, a.k.a. Bézout's little theorem). Since $$P(a)=0$$, the factor theorem tells us that $$P(x)=(x-a)T(x)$$ for some polynomial $$T(x)$$. Now take the derivative of both sides of this equation, using the Product Rule for the right-hand side, and then plug in $$a$$. When you do that, you will see that $$T(a)=0$$ too, so apply the factor theorem again.

(P.S. For the second part, I have nothing to add to the other answer.)