C2-The Factor Theorem and The Remainder Theorem

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  • Created by: evie
  • Created on: 15-05-13 14:37

You can factorise polynomials by using the factor theorem: If f(x) is a polynomial, and f(p)=0, then (x-p) is a factor.


1) Show that (x-2) is a factor of x³+x²-4x-4 by using the factor theorem.

f(2)=2³+2²-(4×2)-4 f(2)=8+4-8-4 f(2)=0

There is no remainder, therefore it is a factor.

2)Factorise 2x³+x²-18x-9

  • we write the polynomial as a function, and try different values of x until we get the answer to be 0. (although it’s not on the spec, the value of x will always be a factor of the last number, in this case, -9)

So the values of x will be -1, 1, -3, 3, -9, or 9

f(1)=2×(1³)+1²-(18×1)-9 f(1)=-24 ~-1…



Amazing thanks so much Evie! this really helped me understand the topic, you've saved my A Level! Make sure you do some more please :D


A good set of worked examples illustrating factor and remainder theorem.  Written in a way that students will understand.  Make sure the statement of results is written clearly at the end of your answers: 

e.g. if f(2) = 0, then x - 2 must be a factor of f(x).  (Factor theorem)

If f(1) = 24 it follows that (x-1) is NOT a factor and gives a remainder of 24 when f(x) is divided by x - 1. (remainder theorem)

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