Algebra : polynomials

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






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:


second term has an index which is not an integer(whole number)


indices of the variable contain integers which are not positive

examples of polynomials:





back to top


Algebraic long division


f(x) the numerator and d(x) the denominator are polynomials


the degree of d(x) <= the degree of f(x)


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.


algebraic long division problem#1


back to top


The Remainder Theorem

If a polynomial f(x) is divided by (x-a), the remainder is f(a).


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.


back to top


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

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


The factor Theorem problem#1




back to top


your stop for the best in math, science & programming tutorials on the Net revision help to get a better result incremental success advanced physics for secondary/high school, including much in-depth content common to first year university courses your one stop for the best in math, science and programming tuition revision help for a better result incremental success advanced physics for high school/secondary and 1st year university fast-track learning for everyone



parametric differen.
equation of a tangent
equation of a normal
rate of change prob.1
rate of change prob.2
the Chain Rule
Chain Rule probs. #1
Chain Rule probs. #2
Chain Rule probs. #3
the Product Rule
parametric eqs.prob#1
parametric eqs.prob#2
intro. to integration
integration by parts 1
integration by parts 2
area under a curve
volumes of revolution
area between curves
Binomial Theorem
Bin. Theorem problems
Trig. Identities
Half Angle Formula
Double Angle Formula
Vector problems
Vector problems in 3D
MORE . . .


derivative formula
tangents & normals
inverse functions
MORE . . .


Edxl C1 Pure specimen
Edxl C1 Pure answers
Edxl C2 Pure specimen
Edxl C2 Pure answers
Edxl C3 Pure specimen
Edxl C3 Pure answers
Edxl C4 Pure specimen
Edxl C4 Pure answers
MORE . . .


the derivative formula
tangents & normals
maxima & minima
chain rule
diffn.trigonometric fns.
product rule
quotient rule
parametric equations
implicit equations
differential equations
integration formula
int.'by substitution'
int.'by parts'
algebraic fractions
definite integrals
areas under curves
volumes of revolution
Trapezium Rule
integ. diff. equations
sine, cosine, tangent
sec, cosec, cotan
Sine & Cosine Rules
Pythagorean ID's
compound angles
Sigma Notation
arithmetic progression
geometic progression
line between points
more straight lines
parametric equations
circles & ellipses
MORE . . .