# Polynomials

A polynomial is defined as an expression of finite length consisting of variables with only positive whole number exponents. For example:

is a polynomial of degree 2.

The degree of a polynomial is the exponent of the leading term of the polynomial.
In the polynomial above, each of the following refers to one term; **
** is one term; **4x**
is another term; and **4** is the last term. The leading term is the term whose
exponent has the highest value and is often the first term of the polynomial. In
the polynomial above, the leading term is **
**. A term is also
referred to as a monomial; which means a polynomial of one term. A term consists
of a variable (in this case **x**) and a constant or coefficient (any number).
This is true despite there being no obvious variable attached to the last term as
shown below:

Keeping in mind that any number to the power zero is one (i.e. **
** = 1), the above
becomes:

Polynomials are commonly denoted by **P(x)**, for example

## Roots of Polynomials

Equating **P(x)** to zero and solving for **x** is referred to as the solving
for the roots of **P(x)**. A root of a polynomial **P(x)** is defined as the
value of **x** for which the polynomial equals zero. Roots of polynomials can
be either positive or negative or zero. Roots can also be real numbers or complex
(imaginary) numbers. If a polynomial has degree **n**, then there must exist
**n** roots of that polynomial. If you were to graph a given polynomial, the
roots of the polynomial would be the points where the curve formed by the polynomial
cuts the x-axis (the x-intercepts).

Given the polynomial below

from the degree of the polynomial (3) we can tell that the polynomial will have 3 roots such that

If the roots of the above polynomial are α, β and γ, then these roots are related by

where each expression within a set of parentheses is known as a factor of the polynomial
(i.e. **(x - α) **is a factor of the polynomial).

Roots of the polynomial **P(x)** can be found by equating the polynomial to zero
and then trying to substitute for different values of** x** to find out which
ones make the equation equal to zero. This is known as factoring.

For example; find the roots of the polynomial

**Step 1**

First step is to equate the polynomial to zero

**Step 2**

next step is to try different values of *x* to find 3 that make the equation
equal to zero

try substituting *x* = 1

this implies that *x* = 1 is a root of the polynomial and (*x* - 1) is
a factor of the polynomial

but the degree of the given polynomial is 3 so we need to find 2 other roots to make a total of 3 roots

**Step 3**

try substituting *x* = -1

this implies that x = -1 is not a root of the polynomial

**Step 4**

try substituting x = 2

this implies that *x* = 2 is a root of the polynomial and (*x* - 2) is
a factor of the polynomial

**Step 5**

try substituting *x* = 3

**Step 6**

Therefore the roots of the polynomial are given by *x* = {1,2,3} and this can
be proved by expanding the factors, i.e.

which is the same polynomial we started with!

If we look at the graph of this function, we can clearly see the roots of our function.

## Quadratic Polynomials

Quadratic Polynomials are polynomials of degree 2; meaning that the leading term has a variable with an exponent of two. Quadratic polynomials are very important in lower level algebra because of the special properties they possess, and thus warrant an entire section of their own under quadratic equations.