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These are important terms and notations for this section.
Monomial A term of the form for some constant and some non-negative integer .
From “mono” meaning “one” and “nomen” meaning “name”.
Binomial An expression that is the sum or difference of two monomials. From ”bi”
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 we could rewrite it in descending order of
exponents, to get which makes clear that as the ‘leading term’ of .
(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 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 .
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: 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 .
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 is a root of the polynomial because . In
comparison is not a root of the polynomial , even though because is not
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: 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
Notation for an arbitrary polynomial The standard notation for a polynomial
() of degree and with coefficients is as follows: This notation should be
explained however. The polynomial , is degree , so its highest power is , and
each of the coefficients “” means that all the coefficients are real numbers.
The fact that there are and the polynomial is degree is not a coincidence;
the subscript on the coefficient and the degree of the polynomial are the
same . Moreover, looking at the definition of above you can see that each
term is of the form meaning that the power of 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 , we would immediately know that is because that’s how they are
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.