Factor & Remainder Theorem (Cambridge O Level Additional Maths)

What is the factor theorem?

• The factor theorem is a useful result concerning the roots and factors of polynomials
  
  • In the example below, the polynomial  has three (linear) factors
    • and 
    • and so it has the three roots  and 

• For a polynomial  the factor theorem states that:

i)

if , then  is a factor of 
( is a root of )

and

ii)

if is a factor of , then

What is the remainder theorem?

• The factor theorem is actually a special case of the more general remainder theorem
• The remainder theorem states that when the polynomial  is divided by  the remainder is 
  • You may see this written formally as
  • In polynomial division
    • would be the result (at the top) of the division (the quotient)
    •  would be the remainder (at the bottom)
    •  is called the divisor
  • In the case when  and hence is a factor of  – the factor theorem!

How do I solve problems involving the remainder theorem?

• If it is the remainder that is of particular interest, the remainder theorem saves the need to carry out polynomial division in full
  • e.g.  The remainder from is 
  • This is because if and
• If the remainder from a polynomial division is known, the remainder theorem can be used to find unknown coefficients in polynomials
  • g. The remainder from is 8 so the value of p can be found by solving , leading to 
  • In harder problems there may be more than one unknown in which case simultaneous equations would need setting up and solving
• The more general version of remainder theorem is if is divided by  then the remainder is 
  • The remainder is still found by evaluating the polynomial at the value of such that  (the divisor is zero) but it is not necessarily an integer