Exercises – Long Day

Exercises – Long Day#

Exercise 1: The Division Algorithm#

Question a#

Determine, using the division algorithm, whether the polynomial \(d(Z)=Z^2+Z+1\) is a factor in the polynomial \(p(Z)=2Z^6-2\).

Answer

Yes, it is a factor since the division algorithm in this case results in a remainder of \(0\). More precisely, we get, using the division algorithm, that

\[2Z^6-2=(Z^2+Z+1)(2Z^4-2Z^3+2Z-2).\]

Question b#

Is \(Z^3+2\) a factor in \(Z^6+2Z^3+8\)?

Answer

No, since the division algorithm gives a non-zero remainder.

Exercise 2: Multiplicity of Roots#

Question a#

We are being informed that \(-2\) is a root of the polynomial \(p(Z)=Z^5+4Z^4+4Z^3+5Z^2+20Z+20\). Determine the multiplicity of the root.

Hint

You can find the definition of multiplicity of roots in Definition 5.6.1 in the textbook.

Hint

In Example 5.6.1 in the textbook a similar problem is treated (in particular the third polynomial in the example).

Answer

The multiplicity is \(2\).

Question b#

Find a fourth-degree polynomial that has the roots \(0\), \(1\), and \(-1\) with multiplicities \(2\), \(1\), and \(1\), respectively.

Hint

Each root \(\lambda\) of a polynomial implies a factor of \(Z-\lambda\) being present in the polynomial according to Lemma 5.6.2. Which factor does a root with a multiplicity of \(2\) imply, according to Definition 5.6.1 in the textbook?

Exercise 3: The Roots and Coefficients of a Polynomial#

Question a#

We are given a cubic (third-degree) polynomial \(p(Z)=Z^3+aZ^2+bZ+c \in \mathbb{R}[Z]\) with \(a,b,\) and \(c\) being some numbers that we do not yet know the values of. We are also informed that the numbers \(10\), \(2\), and \(3\) are roots of \(p(Z)\). Compute \(a\), \(b\), and \(c\).

Answer

\(a=-15\), \(b=56\), and \(c=-60.\)

Exercise 4: A Cubic Polynomial#

We are given the polynomial

\[p(Z)=Z^3+9Z^2+28Z+30.\]

In this exercise we will investigate the roots of this polynomial.

Question a#

Show that \(-3\) is a root of \(p(Z)\).

Question b#

Use the division algorithm to show that

\[p(Z)=(Z+3)(Z^2+6Z+10).\]

Question c#

Now, find all solutions within the complex numbers to the polynomial equation \(p(z)=0.\)

Answer

\(-3,-3+i,-3-i.\)

Exercise 5: Real Roots#

Question a#

Is the following proposition true?

Every polynomial \(p(Z) \in \mathbb{R}[Z]\) of degree four has a real root.

Either prove the proposition true or provide a counterexample to prove it false.

Exercise 6: Factorizing in order to Reduce#

We are informed that \(i\) and \(1+i\) are roots of the polynomial

\[P(Z)=Z^2−Z−2iZ−1+i.\]

Use this info to reduce the following fraction:

\[\frac{Z^2−Z−2iZ−1+i}{Z-1-i}.\]

Exercise 7: Factors with Real Coefficients#

Find all roots in \(\mathbb C\) of the following polynomials and then write the polynomials as a product of polynomials with real coefficients of lowest possible degree.

  1. \(Z^3+Z-2.\)

  2. \(Z^2+2Z+1.\)

  3. \(Z^3-Z^2-3Z+2.\)

Hint

In the proof of Corollary 5.6.4 in the textbook it is explained how this product is found based on the roots of a polynomial.

Answer

  1. The roots are \(1,-\frac12 + \frac12 \sqrt{7}i,\) and \(-\frac12 - \frac12 \sqrt{7}i\). The factorized form is \((Z-1)(Z^2+Z+2)\).

  2. The root is \(-1\) (with multiplicity two). The factorized form is \((Z+1)^2.\)

  3. The roots are \(2,-\frac12 + \frac12 \sqrt{5},\) and \(-\frac12 - \frac12 \sqrt{5}\). The factorized form is \((Z-2)(Z+\frac12 - \frac12 \sqrt{5})(Z+\frac12 + \frac12 \sqrt{5})\).

Exercise 8: Complex Roots and Factors with Real Coefficients#

A polynomial \(p(Z) \in \mathbb{R}[Z]\) of degree four is given in the form \(Z^4+aZ^3+bZ^2+cZ+d\). We are informed that \(p(i)=0\) and \(p(2+i)=0\).

Question a#

Compute all roots of \(p(Z)\).

Hint

Lemma 5.3.3 is useful here.

Answer

\(i,-i,2+i,2-i.\)

Question b#

Write \(p(Z)\) as a product of two quadratic polynomials with real coefficients.

Answer

\((Z^2+1)(Z^2-4Z+5).\)

Question c#

Compute \(a,b,c,\) and \(d\).

Answer

\(a=-4\), \(b=6\), \(c=-4\), and \(d=5.\)

Exercise 9: 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 \(r_2(Z)=11Z+18\). Determine the secret code.

Answer

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

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