In this section we will learn about “roots,” or “solutions,” or how you probably already think of them, x – intercepts.  There are two key ideas we need to understand about x – intercepts, and one is familiar.  We will also discuss the y – intercept.  That’s a little simpler because we are dealing with polynomial functions, which means there will be exactly one y – intercept.

y – Intercept

The y – intercept is where the graph crosses the y – axis.  This is a coordinate and the x value will always be zero.  To find the y – intercept, you simply replace x with zero, and see what’s left.  In function notation, that’s just f(0). 

Let’s see an example.

Find the y – intercept of the polynomial function below.

All we need to do is evaluate f(0).  Let’s do that below.  You’ll see that the value of  is the y  value of the y – coordinate. 

The y – intercept is (0, 14).

x – Intercepts

We often write x – intercepts as coordinates.  Coordinates are an ordered pair, (x, y), where x is the input and y is the output.  For x – intercepts, the y – coordinate is always zero.  That’s why to find x – intercepts, we set the function equal to zero and solve it.  The function notation is f (x) = 0. 

That’s an old path, one we know well for linear equations and quadratic functions.  (All quadratic polynomials are functions, but not all linear equations are functions.  Do you know the exception?) 

Let’s see if you can find the x – intercepts for the following polynomial function.  Note, this function is NOT written in standard form.

Given what we know now, here’s how we find the x – intercepts.

Let’s find the x – intercepts for the function above.

We have four different x – intercepts here.  Do you know the degree of this function?  It is NOT two!  Do you see that each group is multiplying with the other groups.  This will give us a degree of 4.  We can see this pretty easily without multiplying the entire thing together.  We can start by multiplying the binomials together as follows.

Now, if we multiplied those two quadratics together, our largest exponent would be 4.  Our function would start off with 3x⁴. 

Number of x – Intercepts

Let’s introduce this new concept by tying it into something familiar. Consider these two questions.

1. How many x – intercepts can a linear function have?
   
   
   
2. How many x – intercepts can a quadratic equation have?

For question #1, there can be zero x – intercepts or one x – intercept.  If you thought of the special case of y = 0, which is the x – axis, well, you’re quite clever!  But let’s ignore that situation for now.  Does the x – axis intercept itself?  Best to leave that behind for now.

For question #2, there can be zero, one or two x – intercepts. 

Two new questions.

1. What is the degree of linear functions?
   
   
   
2. What is the degree of quadratic equations?

Do you see that linear functions have a degree of one, and a maximum number of solutions of just one.  Similarly, a quadratic equation has a degree of two and a maximum number of solutions of two.  That’s not a coincidence. 

Fact:  The number of x – intercepts cannot exceed the value of the degree.

So, if you have a degree of 21, there could be anywhere from zero to 21 x – intercepts!

The number of solutions will match the degree, always.  However, there can be repeated solutions, as in f(x) = (x – 4)(x – 4)(x – 4).  This function is cubic.  Do you see why?  You can multiply each binomial together, but you can also see the degree without carrying out the entire operation of multiplying.  The image at the top of this section is a graph of this function!