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.