Monomial A term of the form $ax^n$ for some constant $a$ and some non-negative integer $n$. From “mono” meaning “one” and “nomen” meaning “name”.

Binomial An expression that is the sum or difference of two monomials. From ”bi” meaning “two”.

Polynomial A function or expression that is entirely composed of the sum or differences of monomials. From “poly” meaning “many”.

Leading Term (of a polynomial) The 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 $p(x) = x^2 - 13x^3 + 4x - 1$ we could rewrite it in descending order of exponents, to get $p(x) = -13x^3 + x^2 + 4x - 1$ which makes clear that $-13x^3$ as the ‘leading term’ of $p(x)$.

(Complex) Conjugates A pair of complex numbers whose real parts are the same, and whose imaginary parts differ only by a negative sign are called complex conjugates.
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 $5 + 3i$ and $5 - 3i$ are complex conjugates. If one were to ask ‘what is the complex conjugate of $5 - 3i$ the answer would be the other number of the complex conjugate pair, ie $5 + 3i$.

Curvature Curvature refers to monotonicity (increasing/decreasing) and the concavity (bending up or down) of a curve.

Irreducible Polynomial A polynomial that cannot be factored any further. We will often specify under what type of numbers we are factoring the polynomial; eg real numbers or complex numbers. This indicates whether all numbers in the factored form must be real or complex numbers (respectively).
For Example: $(x^2+1)$ is irreducible under the real numbers because there is no way to factor this quadratic with real numbers only. However $x^2+1$ is not irreducible under the complex numbers, as we can write $x^2+1 = (x+i)(x-i)$.

Root (of a polynomial) A root of a polynomial is an irreducible polynomial that is a factor of the given polynomial.
For Example: The polynomial $x + 1$ is a root of the polynomial $x^2 - 1$ because $(x+1)(x-1) = x^2 -1$. In comparison $x^2-1$ is not a root of the polynomial $x^4-1$, even though $(x^2-1)(x^2+1) = x^4-1$ because $x^2-1$ is not irreducible.

Multiplicity (of a value/zero) The multiplicity of a value is a count of how many times that value occurs. This is most often used in reference to the ‘multiplicity of a zero’ or ‘multiplicity of a root’.
For Example: Let’s say we have factored a polynomial into the form: $$p(x) = (x+1)^3(x-1)^2(x+5)(x+17)$$ We would say that “the root $(x+1)$ has multiplicity 3”, because the term $(x+1)$ occurs 3 times (hence the power of 3). Similarly the root $(x-1)$ has multiplicity 2, and the roots $(x+5)$ and $(x+17)$ both have multiplicity 1. This can be even easier to see if we re-write $p(x)$ without using exponents; $$p(x) = (x+1)(x+1)(x+1)\cdot (x-1)(x-1)\cdot (x+5)(x+17)$$ NOTE: When we say that a polynomial has $n$ roots “up to multiplicity” what we mean is that if we add all the multiplicity numbers together of all the roots, we would get $n$. So in the case of $p(x)$ we would say $p(x)$ 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 for an arbitrary polynomial The standard notation for a polynomial ($p(x)$) of degree $n$ and with coefficients $c_0, c_1, \dots , c_n \in \mathbb {R}$ is as follows: $$p(x) = c_nx^n + x_{n-1}x^{n-1} + c_{n-2}x^{n-2} + \dots + c_2x^2 + c_1x + c_0$$ This notation should be explained however. The polynomial $p(x)$, is degree $n$, so its highest power is $n$, and each of the coefficients “$\in \mathbb {R}$” means that all the coefficients are real numbers. The fact that there are $c_0, c_1, \dots c_n$ and the polynomial is degree $n$ is not a coincidence; the subscript on the coefficient and the degree of the polynomial are the same $n$. Moreover, looking at the definition of $p(x)$ above you can see that each term is of the form $$c_{\text {(some value)}} x^{\text {(the same value)}}$$ meaning that the power of $x$ and the subscript on the coefficient match. This is also not an accident, this is how we tell which coefficient goes with which term. For example; if we wanted to know the coefficient of $x^{17}$, we would immediately know that is $c_{17}$ because that’s how they are named.

Finally, notice that the last term in the standard notation is just $c_0$, however, that is better written as $c_0x^0$, meaning that it conforms to all the other terms. It’s just easier to simplify the $x^0 = 1$ and then omit writing it, so we only write $c_0$. But notice that the pattern holds for every term, including the last one, despite the fact that we don’t write the $x^0$ piece of the last term.