Polynomials
Definition
A special family among mathematical functions are polynomials, which are typically noted . They are easily spotted thanks to their beautiful structure. The following example is said to be a polynomial of degree 3:
As the example suggests, polynomials are sums of terms each of which is a constant multiple of a non-negative integer power of the variable . We are now ready for the formal definition:
A polynomial is a function of the following form:
where , and are called coefficients, or constants, of the polynomial. The highest power of in the polynomial is called the degree of the polynomial. (Note that a coefficient can be zero, which eliminates that specific power from the polynomial)
Examples
is a polynomial of degree 0
is a polynomial of degree 1
is a polynomial of degree 3
Factorization
Every polynomial can be factorized, similar to how we are familiar with factorization of numbers, for example the number 21:
However, before we engage in polynomial factorization, lets consider the multiplication of two polynomials, and . The product is in fact a polynomial of degree , if is of degree and is of degree . For example, let's say that and . We have:
Now we are ready for the opposite proposition, that every polynomial can be factorized by their solutions to , also called roots. Let's reuse our example from before:
It's evident from the factorization that has the solutions , and .
Note that not all polynomials have real numbered roots, but if we allow for complex numbers, we have the same number of roots as the degree of the polynomial. They must however not be distinct. For example,
has two distinct roots, and , where is a double root. That gives us:
Polynomial division
Before showing how we perform polynomial division, we will revisit the method used in ordinary long division. The method is quite simple. You start with a number, for example and try to divide it by . The method is to always remove one order of magnitude from the number. The first step is to write:
The next step is to remove . We also put 30 above our equations
After this we find that we got . Next we must remove . We add the upper part with .
After all of this we got our final answer. Note that is remainder in this case. Also note that .
Now we are ready to discuss the polynomial case. To start we define rational functions.
A function is called rational if it can be expressed as
where both and are polynomials.
Sometimes we can simplify these fraction using long division. Dividing polynomials is a more complex procedure but the fundamental idea is the same. Say we want to divide with , both being polynomials. Then we want to find and below
To make this more concrete we will show an example. Say you want to divide with . The first step is to write our polynomial using the same method as above.
Next we focus on finding a way of getting rid of . Note that , because of this we can divide that part of the polynomial away. Note that we get a remainder of 4x.
is not divisible by , but since and therefore we can write: (Again we get a remainder in this case 11).
Therefore
