When you factorize polynomials of degree 3 or higher, you want to use techniques that leave you with a quadratic expression. The most important technique for this is polynomial long division.
When you are left with a quadratic expression, you can factorize that as usual by finding
where and are solutions to the equation
You can find these solutions through inspection, the quadratic formula, or by using digital tools. But first, let’s take a closer look at polynomial long division:
Theory
If , then is a solution to the equation . That means the division is solvable and has no remainders.
When is solvable without a remainder, you get , where is a new polynomial of a lower degree than .
If is of degree 3 or higher, you have to find more solutions. Solutions to are also solutions to . If is of degree 2, you can factorize the expression like you normally would.
is a factorization of .
In exercises where you are factorizing polynomials of higher degrees, you are either given a value that you are supposed to check, or there is a clear hint about what is correct. If not, one of the solutions will often be or . You check the solution by inserting the value of into the polynomial to see if the answer is 0. If it is, you know that the value you inserted is a solution to the equation, and you can start factorizing.
Rule
If not, go to Item 2.
Here are some examples of factorization of polynomials of degree :
th-degree polynomials have a maximum of solutions, but they can also have fewer. For example, a cubic polynomial has either one, two, or three real solutions. In cases with only one real solution, the factorization looks like this:
Here is the only real solution to the cubic polynomial. The entire graph of the function lies above the -axis, which means it doesn’t produce any solutions.
Example 1
Factorize the polynomial
Because is not present in all of the terms, you have to guess at solutions. Let’s start with :
You continue by trying :
This tells you that divides . That means you can perform the polynomial long division
You have found an expression of degree 2, just like you wanted. You can now factorize this like you always have, either with the quadratic formula, or through inspection.
Finding and :
which gives you and . That means the factorization of the quadratic expression is .
Now, you can set up the finished factorization of the cubic expression, which is
Example 2
Solve the equation
To solve equations like these, you begin by moving everything around to get zero on the right-hand side of the equation. In this case, that has already been done. Next, you factorize the expression on the left-hand side. As this is the same polynomial you had in Example 1, it is the following: