Introduction
A polynomial is an expression which:
- consists of a sum of a finite number of terms
- has terms of the form kxn
(x a variable, k a constant, n a positive integer)
Every polynomial in one variable (eg 'x') is equivalent to a polynomial with the form:
Polynomials are often described by their degree of order. This is the highest index of the variable in the expression.
eg: containing x5 order 5, containing x7 order 7 etc.
These are NOT polynomials:
3x2+x1/2+x
second term has an index which is not an integer(whole number)
5x-2+2x-3+x-5
indices of the variable contain integers which are not positive
examples of polynomials:
x5+5x2+2x+3
(x7+4x2)(3x-2)
x+2x2-5x3+x4-2x5+7x6
Algebraic long division
If
f(x) the numerator and d(x) the denominator are polynomials
and
the degree of d(x) <= the degree of f(x)
and
d(x) does not =0
then two unique polynomials q(x) the quotient and r(x) the remainder exist, so that:
Note - the degree of r(x) < the degree of d(x).
We say that d(x) divides evenly into f(x) when r(x)=0.
Example
The Remainder Theorem
If a polynomial f(x) is divided by (x-a), the remainder is f(a).
Example
Find the remainder when (2x3+3x+x) is divided by (x+4).
The reader may wish to verify this answer by using algebraic division.
The Factor Theorem
( a special case of the Remainder Theorem)
(x−a) is a factor of the polynomial f(x) if f(a) = 0
