Factor & Remainder Theorems (Edexcel IGCSE Further Pure Maths): Revision Note

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  

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  

  • 

  • 

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

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 

  •