Extra (optional) exercises – Long Day

These exercises are optional extra material for students who have finished the exercises of the day and have the time and interest for more.

Extra exercise 1: Multiplicity and derivatives

For a given polynomial \(p(Z)=a_0+a_1Z+a_2Z^2+\cdots+a_nZ^n\) 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 at least two.

Question a

Assume that \(\lambda \in \mathbb{C}\) is a root of \(p(Z)\) with a multiplicity of at least 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 at most 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 \(d_1(Z)=Z^2+1\) using the division algorithm. Another share, the remainder \(r_2(Z)\) from dividing \(p(Z)\) by \(d_2(Z)=Z^2-1\) using the division algorithm, is given to person B.

Question a

Person A is now 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 \(d_1(Z)=Z^2+1\) 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 \(p(Z)=2Z^3+10Z^2+9Z+8\)).