Polynomial
A polynomail is a function in one or more variables that consists of a sum of variables raised to powers and multiplied by coefficients.
For example, these are polynomials:
These aren't polynomails:
Contents
Introductory Topics
A More Precise Definition
A polynomail in one variable, is a function . Here,
is the
th coefficient, and
is an integer.
Finding Roots of Polynomails
What is a root?
A root is a value for a variable that will make the polynomail equal zero. For an example, 2 is a root of because
. For some polynomails, you can easily set the polynomail equal to zero and solve the equations to find roots, but in some cases it is much more complicated.
The Fundamental Theorem of Algebra
The fundamental theorem of algebra states that any polynomail can be written as:
where
is a constant, and
is the highest power of
that
contains (also called the degree). It's very easy to find the roots of a polynomail in this form, because the roots will be
. This also tells us that a polynomail can have up to
distinct roots, where
is its degree.
Factoring
Different methods of factoring can help find roots of polynomails. Consider this polynomail:
This polynomail easily factors to:
Now, the roots of the polynomail are clearly -3, -2, and 2.
The Rational Root Theorem
Descartes' Law of Signs
Binomial Theorem
Binomail theorem can be very useful for factoring and expanding polynomails.
Intermediate Topics
Multiplying and Dividing Polynomials
Synthetic Division
Intermediate and Olympiad Topics
Transforming Polynomails
Other Important Topics
Other Resources
An extensive coverage of this topic is given in A Few Elementary Properties of Polynomials by Adeel Khan.