Unit 2 - Functions
Required Prior Knowledge
Questions
What is the conjugate of the complex number \(3-5i\)?
If\(P\left(x\right)=x^{2}-4x+5\), evaluate \(P\left(2+i\right)\).
Solutions
Get Ready
Questions
How many \(x\)-intercepts can the graph of a polynomial of degree \(n\) have?
How many stationary points can a polynomial of degree \(n\) have?
Solutions
Notes
The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has exactly \(n\) roots, some of which may be repeated, within the complex numbers.
That is, if $$f\left(x\right)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_{1}x+a_{0}$$where \(a_{n},a_{n-1},\cdots,a_{1},a_{0}\in\mathbb{R}\), then $$f\left(x\right)=a_{n}\left(x-\omega_{1}\right)\left(x-\omega_{2}\right)\cdots\left(x-\omega_{n}\right)$$for \(\omega_{k}\in\mathbb{C}\).
The Conjugate Root Theorem states that, for a polynomial with real coefficients, if \(z=a+bi\in\mathbb{C}\) is a root then so is \(z^{*}=a-bi\).
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Consider the polynomial \(f\left(x\right)\) of degree \(n\), with all real coefficients, such that \(f\left(z\right)=0\) for \(z\in\mathbb{C}\).
Then$$\begin{align}f\left(z^{*}\right)&=a_{n}\left(z^{*}\right)^{n}+a_{n-1}\left(z^{*}\right)^{n-1}+\cdots +a_{1}\left(z^{*}\right)+a_{0}\\&=a_{n}\left(z^{n}\right)^{*}+a_{n-1}\left(z^{n-1}\right)^{*}+\cdots +a_{1}\left(z\right)^{*}+a_{0}\\&=\left(a_{n}z^{n}\right)^{*}+\left(a_{n-1}z^{n-1}\right)^{*}+\cdots +\left(a_{1}z\right)^{*}+a_{0}\\&=\left(a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{1}z+a_{0}\right)^{*}\\&=\left(f\left(z\right)\right)^{*}\\&=0^{*}\\&=0$$Thus \(z^{*}\) is a root of \(f\left(x\right)\).
These two theorems, The Fundamental Theorem of Algebra (FTA) and The Conjugate Root Theorem (CRT) are incredibly important results that allow us to determine roots of polynomials of higher degree than 2.
Two other important corollaries of the Fundamental Theorem of Algebra are (can you explain why these are true?):
Every real polynomial can be expressed as a product of real linear and irreducible quadratic factors;
Every real polynomial of odd degree has at least one real root.
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Earlier in your Mathematical journey you will have met the Fundamental Theorem of Arithmetic, though probably not by that name. This theorem states that every positive integer greater than 1 can be written uniquely as a product of primes.
There are two cases for a given integer \(k\):
\(k\) is prime and thus can be written as \(k\) which is a (rather boring) product of a single prime;
\(k\) is composite and can be written in the form $$k=p_{1}^{n_{1}}\times p_{2}^{n_{2}}\times\cdots p_{w}^{n_{w}}$$ where \(p_{j}\) are all primes raised to some power.
For example$$1200=2^{4}\times 3\times 5^{2}$$We could ‘expand’ these powers to \(1200=2\times 2\times 2\times 2\times 3\times 5\times 5\) but this is the same calculation. If we multiply four 2s, one 3 and two 5s, we will always get \(1200\).
The third and final fundamental theorem you will meet in this course is the Fundamental Theorem of Calculus. This will be introduced in Unit 5.
Examples and Your Turns
Example
Given that \(4+5i\) is a complex root of the polynomial \(f\left(x\right)=x^{3}-6x^{2}+25x+82\), find all the remaining roots.
Your Turn
Given that \(2+i\) is a complex root of the polynomial \(f\left(x\right)=x^{3}-7x^{2}+17x-15\), find all the remaining roots.
Your Turn
Given that \(1-i\sqrt{6}\) is a complex root of the polynomial \(f\left(x\right)=x^{3}+x^{2}+x+21\), find all the remaining roots.
Your Turn
Given that \(i\) is a complex root of the polynomial \(f\left(x\right)=x^{4}-2x^{3}+6x^{2}+ax+5\) where \(a\in\mathbb{R}\), find the value of \(a\).
Hence, or otherwise, find the remaining roots.
Your Turn
Given that \(P\left(x\right)=x^{4}-2x^{3}-x^{2}+ax+b\), where \(a,b\in\mathbb{R}\), has \(1+2i\) as a root, find the value of \(a\) and \(b\).
Hence, or otherwise, find the remaining roots.
Your Turn
Solve the equation$$z^{5}-6z^{3}-2z^{2}+17z-10=0$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
If a cubic polynomial has real coefficients, how many real roots can it have? Why?
Describe what a repeated root is, and how these might show up in terms of the Fundamental Theorem of Algebra.
Common Mistakes / Misconceptions
Confusing the domain and range of a relation.
Thinking that all equations represent a function (e.g. \(x^{2}+y^{2}=1\)).
Misunderstanding the vertical line test.
Connecting This to Other Skills
This skill is fundamentally important for the development of all other function-related skills covered in Unit 2.
A clear understanding of relations and functions is absolutely essential for success in calculus, which is explored in greater detail in Units 4 and 5.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?