Written by tutor Eric J.
Overview of the Factor Theorem
The Factor Theorem is an algebraic topic that involves finding the roots (or zeros) of a polynomial function. There are two methods
that one can use in discovering the roots:
- Trial and Error Method
- Berry Method (for binomial functions only)
The definition for factor theorem is for a function, f(x) and when f(s)=0 then (x-s) is a factor of the polynomial. The other factors
can be found using long division or synthetic division once (x-s) has been established.
All polynomials have the same basic form, the only difference is what degree they are i.e. 2nd degree, 3rd degree,…,
to nth degree.
- Example 1 – Ax2 + Bx + C (2nd degree or binomial)
- Example 2 – Ax3 + Bx2 + Cx + D (3rd degree or trinomial)
Note- Where A,B,C,D are constants
Factoring into Binomials
Factoring a polynomial of the 2nd degree into binomials is the most basic concept of the Factor Theorem. For example, given the polynomial
f(x) = x2 + 6x + 5, the factors are (x+1) and (x+5), which make the roots or zeros of the function -1 and -5. Let’s take a look at how I arrived
to this conclusion:
With the function given above you can start always on your scratch paper by writing down (x+?)(x+?) this will be the same for all functions where
the constant in front of the x2 term is 1. The key is finding what the question marks are. This can be achieved by using the Berry Method. The
question I would ask myself is what two same numbers add to be 6 and multiply to be 5 (Constants B and C).
Incorporating the Berry Method
As you can see in the above diagram I placed 5 and 6 in the upper and lower part of the “X”. And the 1 and 5 in the left and right part of the “X”,
and as you can see 5 and 1 do add to be 6 and 5 and 1 do multiply to be 5. So the roots are -5 and -1. This diagram can be used in any other similar
examples so a student can easily evaluate the roots of the binomial. Let’s take a look at one more example…
Suppose you see the function f(x) = x2 – 14x + 45 and you are asked to find the roots. What is the correct answer? You can use the Berry
method diagram again if needed or just ask yourself what two same numbers add to be negative 14 and multiply to be positive 45. In this case it is
negative 5 and negative 9. So the roots would be positive 5 and positive 9.
(x-5)(x-9) = 0, where x equals 5 and 9.
Factoring into Trinomials
Factoring a polynomial into a set of trinomials can be a bit more difficult but can also be easily achieved as well. In this case, you will have
to use a trial and error method to find a factor (x-s). Suppose you have the function f(x) = x3 – 6x2 + 11x – 6 and you
need to find the zeros of the function. The zeros turn out to be 1, 2, and 3. Let’s see how we got to these answers:
Trial and Error Method
With the trial and error method you will need to try and guess a number that makes the polynomial equal to zero. This sounds like a time consuming
task but most algebra texts will have a zero of a function between negative 5 and positive 5. Say you try plugging in 4 into the above equation.
If you do, you will get an answer of f(x) = 6 which is not equal to zero so 4 cannot be a root. But say you plug in 1, now you do get f(x) = 0 so 1 is
a root so you can conclude that (x-1) is indeed a factor of the polynomial.
Using Synthetic Division to find other factors
Now that we have established that (x-1) is one factor of the trinomial you can use synthetic division to find the other two factors.
So by synthetic division another factor of the trinomial is (x2 – 5x + 6) which can further be factored into
(x-2)(x-3) (See above notes on factoring into binomials.)
So now f(x) can be written as f(x) = (x-1)(x-2)(x-3). So to find the roots you set each of these factors equal to zero.
(x-1)(x-2)(x-3) = 0, where x equals 1, 2, and 3.
Factoring polynomials can seem difficult at first but with practice it become easy and second nature. The main thing is to practice similar examples
to the ones I have given in this help section to gain confidence. Always remember practice makes perfect.
Factor Theorem Quiz
True or false? When you factor a 3rd degree polynomial it factors into trinomials.
True or false? If given a polynomial function f(x) and f(s)=1, then (x-s) is a factor of the polynomial.
True or false? The trial and error method is used in factoring a polynomial of the 3rd degree.
True or false? You cannot use synthetic division to find other factors of a 3rd degree polynomial.
True or false? Practice makes perfect.