Extra (Optional) Exercises – Long Day

These exercises are optional extra material for students who are done with the exercises of the day but have the time for and the interest in more.

Extra Exercise 1: Multiplicity and Derivatives

For a given polynomial \(p(Z)=a_0+a_1Z+a_2Z^2+\cdot+a_nZ^n\) we define its derivative \(p'(Z)\) as usual:

\[p'(Z)=a_1+2a_2Z+\cdots+na_nZ^{n-1}.\]

In this exercise we will assume that \(p(Z) \in \mathbb{C}[Z]\) is a polynomial of degree no smaller than two.

Question a

Assume that \(\lambda \in \mathbb{C}\) is a root of \(p(Z)\) with a multiplicity of two. Show that we then have \(p(\lambda)=0\) and \(p'(\lambda)=0\).

Hint

If \(\lambda \in \mathbb{C}\) is a root of \(p(Z)\) with a multiplicity of at least two, then a polynomial \(q(Z)\) exists such that \(p(Z)=(Z-\lambda)^2\cdot q(Z)\). What does this formula imply for the derivative of \(p(Z)\)?

Question b

Assume that \(p(\lambda)=0\) and \(p'(\lambda)=0\) for some complex number \(\lambda\). Show that \(\lambda\) then is a root of \(p(Z)\) with a multiplicity of at least two.

Question c

Conclude that \(\lambda\) is a root of \(p(Z)\) with a multiplicity of at least two if and only if \(p(\lambda)=0\) and \(p'(\lambda)=0\). Also conclude that \(\lambda\) is a root of \(p(Z)\) with multiplicity one if and only if \(p(\lambda)=0\) and \(p'(\lambda) \neq 0\).

Extra Exercise 2: Secret Sharing and the Division Algorithm

We are tasked with uncovering a secret code, known to be the value of \(p(10)\), where \(p(Z)\) is a specific polynomial of degree no higher than three with integer coefficients. A so-called share of this code is given to person A, defined as the remainder \(r_1(Z)\) from dividing \(p(Z)\) by \(d(Z)=Z^2+1\) using the division algorithm. Another share is given to person B, which is the remainder \(r_2(Z)\) from dividing \(p(Z)\) by \(d(Z)=Z^2-1\) using the division algorithm.

Question a

Person A is now being told that their share is \(r_1(Z)=7Z-2\). Provide two different polynomials \(p_1(Z)\) and \(p_2(Z)\) with integer coefficients and degrees no higher than three that both yield a remainder of \(7Z-2\) when divided by \(d(Z)=Z^2+1\) using the division algorithm. Conclude that person A cannot uncover the secret code on his/her own. (A similar argument shows that person B also cannot crack the secret code alone).

Question b

Now persons A and B meet and reveal their shares to each other. The shares are \(r_1(Z)=7Z-2\) and \(r_2(Z)=11Z+18\), respectively. Determine the secret code.

Answer

The code is \(3098\) (and \(p(Z)=2Z^3+10Z^2+9Z+8\)).