These are important terms and notations for this section.

#### Terminology

*leading term*of a polynomial is the term with the largest exponent, along with its coefficient. Another way to describe it (which is where this term gets its name) is that; if we arrange the polynomial from highest to lowest power, than the first term is the so-called ‘leading term’.

**For Example:**For the polynomial we could rewrite it in descending order of exponents, to get which makes clear that as the ‘leading term’ of .

**Note:**We often ask for ‘the complex conjugate to’ a complex number, in which case we are asking for the associated number in the pair.

**For Example:**The numbers and are complex conjugates. If one were to ask ‘what is the complex conjugate of the answer would be the other number of the complex conjugate pair, ie .

**For Example:**is

*irreducible under the real numbers*because there is no way to factor this quadratic with real numbers only.

**However**is

*not*irreducible under the complex numbers, as we can write .

**For Example:**The polynomial is a root of the polynomial because . In comparison is

*not*a root of the polynomial , even though because is

*not*irreducible.

**For Example:**Let’s say we have factored a polynomial into the form: We would say that “the root has multiplicity 3”, because the term occurs 3 times (hence the power of 3). Similarly the root has multiplicity 2, and the roots and both have multiplicity 1. This can be even easier to see if we re-write without using exponents;

**NOTE:**When we say that a polynomial has roots “up to multiplicity” what we mean is that if we add all the multiplicity numbers together of all the roots, we would get . So in the case of we would say has 7 roots “up to multiplicity” since there are 4

*unique*“roots”, but two of them occur more than once, so there are a total of 7 roots if you account for repeats.

#### Notation

Finally, notice that the last term in the standard notation is just , however, that is better written as , meaning that it conforms to all the other terms. It’s just easier to simplify the and then omit writing it, so we only write . But notice that the pattern holds for every term, including the last one, despite the fact that we don’t write the piece of the last term.