Polynomials & their Properties
Irrationals Revisited
Irrational numbers such as $\sqrt{2}$ can be seen as solutions to particular identities.
Polynomials & Their Roots
In general, a polynomial of degree $k$ in $x$ is defined through $k+1$ coefficients:
\[(c_0,c_1,\ldots,c_k)\]This can be written in the following functional form:
\[p(x)=\sum^k_{n=0}c_nx^n\]The coefficient, $c_t$, multiplies $x^t$ when $p(x)$ is evaluated at some $x=\alpha$. This is written as $p(\alpha)$.
Coefficients Expanded
The number types used to choose coefficients can be from any of the sets we have seen so far.
Coefficients can also be limited to finite subset of these:
- $\Bbb{Z}_m$
- The integers $\{0,1,2,\ldots,m-1\}$ with modulo $m$ arithmetic.
If $H$ is such a set $H_k[X]$ or $H[X]$, is used for the set of (degree $k$) polynomials with coefficients in $H$.
Roots of Polynomials
These are values that can be assigned to $x$ that make the function evaluate to 0.
- A polynomial of degree $k$ in $x$ has $k$ roots.
- $(r_1,r_2,\ldots,r_k)$
- These are not necessarily distinct, hence the rounded brackets.
- It is also possible that some are not real numbers.
Some examples of simple polynomials were given along with their roots and degrees.
Operations
- Scalar multiplication is when you multiply each $c_k$ by $\alpha$.
We are expected to know how to add, multiply and divide polynomials.
Finding Roots
Finding roots for polynomials can be very useful for optimisation, using spectral methods for image compression and image compression.
Completing these tasks programmatically can be very taxing, especially for large polynomials.
We can give a finite description of an irrational number as the root of a minimal polynomial.
It is not possible to define all real numbers as the root of a polynomial with only rational coefficients $Q[X]$. A famous example of this is $\pi$.