What are polynomial factors?
The factors of any number are simply the numbers that multiply together to equal that number. For instance, 4 is a factor of 24, since 4 × 6 = 24.
For polynomials, factors are typically other polynomials! A polynomial could look like x2 - x - 6, which has two factors.
The two factors are (x - 3) and (x + 2). We can verify this by multiplying these two factors together, using the FOIL method.
The FOIL method is a mnemonic device to remember how to multiply binomials together. It stands for
To multiply (x - 3)(x + 2), multiply the first of each binomial to find x × x = x2 .
Combine the middle two terms and add all of this together to find the polynomial x2 - x - 6.
Factoring by itself is a very useful skill, but this lesson will focus more on what we can use the polynomial factors to do.
Most importantly, polynomial factors tell us the x-intercepts, zeros, or roots of our polynomial. All three of these names mean the same thing: where our polynomial crosses the x-axis.
In the previous example we found that x2 - x - 6 = (x - 3)(x + 2). Since we are interested in the graph of this polynomial, we can set it equal to and see the following graph.
Notice that this quadratic graph crosses the x-axis at (-2, 0) and (3, 0).
This seems a bit odd as the polynomial factors (x - 3) and (x + 2). It feels a bit like these x-intercepts should be -3 and positive 2.
What’s happening here is that since we are considering the graph y = (x-3)(x+3), we are trying to determine where y = 0. Plugging this into the equation, we see that 0 = (x-3)(x+3).
This means that one of two things is true. Either x - 3 = 0 or x + 2 = 0. Take a moment and consider why that is.
Notice that since the left hand side of the equation is 0, we know one of the two factors must be 0 on the right hand side.
Solve both of these equations for x to find that x = 3 or x = -2, which is exactly what we discovered from the graph alone!
This is why polynomial factors are so powerful. They immediately show us where the zeroes of our polynomial are.
If I told you that some polynomial p(x) had a factor of 4x + 3, you wouldn’t be able to tell me much about the polynomial, but we would know that when 4x + 3 = 0 we could find that x = -3/4 and know that this is one of the -intercepts of our graph!
The other major use of polynomial factors is determining the behavior of our graph near the x-axis.
For example, the graph of y = (x - 2)2(x - 2)3(x + 1) looks something like this:
Notice that all of the x-intercepts are -1, 1 and 2, just like we would predict.
However, we can be more specific with how the graph is behaving at each of these points.
The x-intercept at -1 is the most straightforward. Since this corresponds to the polynomial factor (x + 1) , which is not raised to any power, the graph simply crosses the x-axis at x = -1.
The x-intercept at 2 is the second most straightforward. Notice that this corresponds to the polynomial factor (x - 2)2. Since this polynomial factor is squared, the graph acts like a parabola and only touches our x-axis, not crossing it!
This is what will happen for every polynomial factor that is squared.
Finally, we can consider the x-intercept at 1. This corresponds to the polynomial factor (x - 1)3 and it crosses the x-axis, but it behaves a bit strangely. Notice that the graph initially looks like it will not cross, but then it does cross the x-axis.
This is because this is how the graph of y = x3 behaves as well! When you see this behavior, it is due to a cubed polynomial factor.
Ultimately, polynomial factors are a powerful and versatile tool that allow you to understand the zeros of polynomials and their graphs in much greater detail.