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The Factor and Remainder theorem
When we are given a function in the exam, f(x), normally a cubic, and asked to find a factor, verify a
factor, find a remainder, find the quotient, find the roots or sketch the function, we often have to use
the factor theorem and/or the remainder theorem.
The factor theorem
If (x+a) is a factor of f(x) then f(-a) = 0.
What this means in practice is that to verify that (x+a) is a factor, we simply substitute (-a) into x, and
then simplify to show that this equals zero. If it does not equal zero, then (x+a) cannot be a factor.
Uses of this:
If we are trying to solve a cubic or find the roots or sketch it, then we can use the factor
theorem to help us find the first factor before we equate coefficients.
Sometimes the exam question just says: “verify that .. is a factor of f(x)”, in which case you
simply state the theorem and then show the substitution.
Sometimes you are given f(x) where one of the terms is “ax” or “kx” and then a factor and asked
to find a or k. You can just state the factor theorem, substitute everything in that you can and
you end up with a linear equation which you can solve to find “a” or “k”.
An example
x3-7x-k
(x+1) is a factor.
If (x+1) is a factor then f(-1)=0
(-1)3-7(-1)-k=0
-1+7-k=0
6-k=0
k=6
The Remainder theorem
If (x+b) is not a factor of f(x), then f(-b) is the remainder when f(x) is divided by (x+b)
This means that if we want to find the remainder of a polynomial divided by a factor, we need only find
f(-b) to find this remainder. It is only if we also need the quotient that we must use equating
polynomials. This rule can therefore save a lot of time in the exam!
Similar questions to the one above are popular with exam boards. They give you the remainder, when
f(x) is divided by (x +b) and then ask you to find a constant within f(x). A similar method must be
followed, forming a linear equation which can be solved.
An example
The polynomial x3+3x2+ax+b has a remainder of 3 when it is divided by x+1 and a remainder of 15 when
it is divided by x-2. Find the remainder when it is divided by x+3.
There are two unknowns that we need to find out first, a and b. As we have 2 bits of information about
remainders, we can assume that we need to form two simultaneous equations.
Start with the first bit of info: a remainder of 3 when divided by (x+1)
This means f(-1) = 3
(-1)3+3(-1)2+a(-1)+b=3
-1+3-a+b=3
-a+b=1
Now the other information we have is a remainder of fifteen when divided by (x-2)
So, f(2) = 15
23+3(2)2+2a+b=15
Simplifies to
2a+b=-5
So now we can subtract –a+b =1 from 2a+b=-5 to give a=-2
Substitute this back into one of the equations to find b=-1
Now put these values back into the original polynomial to get x3+3x2-2x-1.
To finish the question we need to find the remainder when it is divided by (x+3). So we simply find f(-3),
showing working, and we get 5. This is the answer.
The factor and remainder theorem is relatively simple maths. This means that the exam is likely to make
you try and apply it in a novel way, combining it with something else. Break things into simple stages,
perhaps write both theorems out once you realise that they are what the question is about.