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
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#
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
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
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#
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: Factorising in order to Reduce#
We are informed that \(i\) and \(1+i\) are roots of the polynomial
Use this info to reduce the following fraction:
Exercise 7: Factors with Real Coefficients#
Find all roots in \(\mathbb C\) of the polynomials in the following questions and then write the polynomials as a product of polynomials with real coefficients of lowest possible degree.
Question a#
\(Z^3+Z-2.\)
Hint
In the proof of Corollary 5.6.4 in the textbook it is explained how such product is found from the roots of the polynomial.
Answer
The roots are \(1,-\frac12 + \frac12 \sqrt{7}i,\) and \(-\frac12 - \frac12 \sqrt{7}i\).
The factorised form is \((Z-1)(Z^2+Z+2)\).
Question b#
\(Z^2+2Z+1.\)
Answer
The root is \(-1\) (with multiplicity two).
The factorised form is \((Z+1)^2.\)
Question b#
\(Z^3-Z^2-3Z+2.\)
Answer
The roots are \(2,-\frac12 + \frac12 \sqrt{5},\) and \(-\frac12 - \frac12 \sqrt{5}\).
The factorised 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 \(d=5.\)
Are you Done with all Exercises?#
If it went too quick and you are eager for more then click here for some challenging optional extra exercises.