Factor & Remainder Theorems (Edexcel IGCSE Further Pure Maths)
Revision Note
Written by: Roger B
Reviewed by: Dan Finlay
Factor Theorem
What is the factor theorem?
The factor theorem is used to find the linear factors of a function
This is closely related to finding the roots (or solutions) of a function or equation
For a function , the factor theorem tells us that
If , then is a factor of
If is a factor of , then
How do I use the factor theorem?
Consider the function where is a factor
Then by the factor theorem we know that
I.e., is a solution to the equation
Or consider the function where
Then by the factor theorem we know that is a factor of
Therefore
where is a function that is also a factor of
Hence
I.e. is the quotient when is divided by
And the remainder is equal to zero
If the linear factor has a coefficient of x (other than 1) you must first factorise out the coefficient
For the linear factor
Examiner Tips and Tricks
Be careful with the minus sign in a factor
That means is a solution to , not !
If you are looking for integer solutions to (where is a polynomial)
those solutions will always be factors of the constant term in
Worked Example
a) Consider the function. Given that is a solution to the equation , write down a linear factor of .
By the factor theorem, if then is a factor of
b) Use the factor theorem to determine whether is a factor of .
By the factor theorem, can only be a factor of if .
But be careful – here is equal to , not
, so is not a factor of
c) It is given that is a factor of . Find the value of .
, so is a factor of .
Therefore by the factor theorem, .
Remainder Theorem
What is the remainder theorem?
The remainder theorem is used to find the remainder when we divide a polynomial function by a linear function
When a polynomial function is divided by a linear function , the value of the remainder is given by
Note, if then is a factor of ; this is the factor theorem
How do I use the remainder theorem?
Consider a polynomial function and a linear function
is the quotient (also a polynomial function)
is the remainder (a real number)
This may also be written as
The remainder theorem tells us that
I.e. we don't need to do the algebraic division to find the remainder!
If the linear factor has a coefficient of x (other than 1) then you must first factorise out the coefficient
For the linear function
Examiner Tips and Tricks
Be careful with the minus sign in
You need to put into to find the remainder, not !
Worked Example
a) Find the remainder when the function is divided by .
We're dividing by
So
By the remainder theorem the remainder will be
Remainder = 7
b) The remainder when is divided by is 3. Find the value of .
So here the value of to use is
By the remainder theorem the remainder will be equal to
Last updated:
You've read 0 of your 5 free revision notes this week
Sign up now. It’s free!
Did this page help you?