Factoring Higher Degree Polynomial Functions

In order to sketch the graph of a higher-degree polynomial function, you must first be able to factor it. Factoring a higher degree polynomial function relys on the Factor Theorem.

Factor Theorem

The factor theorem states:

For any given polynomial, if f(a) = 0, then (x - a) is a factor of f(x)

You must use this rule when attempting to factor ANY higher-degree polynomial. How it is used is shown below:

Example 1

Factor the polynomial function .

Solution

In order to factor this polynomial, we need to discover at least one of it's factors. The tricks you learned when you were factoring trinomials don't work here anymore! The way to discover ONE of the factors of the above polynomial is by using the factor theorem.

The way the factor theorem is used is that you guess a value for x that might make the polynomial equal to zero (this is your a value). Then substitute that value for x and test.

Let's go through this process now. Let's try an a value of 0:

If a = 0, then:

This does not result in a function value of zero, so let's pick ANOTHER value for x. Let's try 1:

If a = 1, then:

We know that substituting an a value of 1 makes the value of the function equal to zero. Using the factor theorem:

For any given polynomial, if f(a) = 0, then (x - a) is a factor of f(x)

we know that f(1) = 0, and therefore (x - 1) is a factor of .

Knowing this fact allows us to divide the factor (x - 1) out of f(x). If you need help with polynomial long division, see that section in the Polynomials Unit of the Algebra B course. The division is shown below in its entirety:

What this means is that when we factor (x - 1) out of x³ + 2x² - x - 2, we get a result of x² + 3x + 2. Now we can re-write our polynomial in a partially factored form:

But this is not quite complete. We can still factor the trinomial in the second set of brackets. (It factors to (x+2)(x+1)). Therefore, our final factored form is:

This is just an introduction to factoring: make sure you have all your notes for this topic - and practice! You will need this skill in order to graph higher degree polynomial functions. You can now head back to the higher degree polynomial functions main page.