,
Algebra : polynomials
 

[ intro ][ algebraic long div.][ Remainder Th. ][ Factor Th. ]

 

 

 

 

Introduction

A polynomial is an expression which:

  1. consists of a sum of a finite number of terms
  2. 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:

polynomial structure

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

 

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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:

polynomials #2

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

algebraic long division problem#1

 

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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).

Remainder Theorem problem

The reader may wish to verify this answer by using algebraic division.

 

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The Factor Theorem
( a special case of the Remainder Theorem)

(xa) is a factor of the polynomial f(x) if f(a) = 0

Example

The factor Theorem problem#1

 

 

 

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