Exercises – Long Day#
Exercise 1: The division algorithm#
Question a#
Use the division algorithm to determine whether the polynomial \(d(Z)=Z^2+Z+1\) is a factor of 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, the division algorithm gives
Question b#
Is \(Z^3+2\) a factor of \(Z^6+2Z^3+8\)?
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 the 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
According to Lemma 5.6.2, each root \(\lambda\) of a polynomial means that \(Z-\lambda\) is a factor of the polynomial. Which factor does a root of multiplicity \(2\) give, 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 \(p(Z)=Z^3+aZ^2+bZ+c \in \mathbb{R}[Z]\), where \(a\), \(b\) and \(c\) are unknown numbers. 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 complex solutions 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 present a counterexample to show that it is false.
Exercise 6: Factorising to simplify#
We are informed that \(i\) and \(1+i\) are roots of the polynomial
Use this information to simplify the following fraction:
Exercise 7: Factors with real coefficients#
Find all roots in \(\mathbb C\) of the polynomials below, and then write each polynomial as a product of polynomials with real coefficients of the 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 a 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 \(2\)).
The factorised form is \((Z+1)^2\).
Question c#
\(Z^3-Z^2-3Z+2.\)
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 \(d=5\).
Have you finished all the exercises?#
If it went too quickly and you are eager for more, then click here for some challenging optional extra exercises.