How Many Times Does It Cross?

(An Introduction to the Fundamental Theorem of Algebra)

Objective: To investigate the graphs of polynomial functions to see a connection between the number of roots of a graph of a polynomial and the degree of the polynomial.

Exercises

1. Let us look at the graph of the polynomial . Does your graph match the graph from Graphing Calculator that I have below?

1. Looking at the polynomial only, what is the degree of the polynomial?

2. Now looking at the graph, how many times does the graph cross the x-axis?

3. Specifically, where does it cross the x-axis?

4. Do you see a connection between your answer to (a) and (b)? If so, what is it?

2. Let’s try a different polynomial and see if the same thing happens. The polynomial this time is .
1. Sketch the graph of the polynomial below. The shape of the graph is no so important as how and where it crosses the x-axis.

2. How many times does the graph of the polynomial cross the x-axis?

3. Where does the graph gross the x-axis?

4. Looking back at the polynomial, what is the degree of f(x)?

5. What is the connection between the answers in (b) and (d)? Is it the same as the connection you made to Exercise #1?

3. Let us test our conjecture on the polynomial . Does your conjecture still hold from Exercises #1 and #2?

4. Write down your conjecture so far. Relate the number of roots (the number of times a graph crosses the x-axis) to the degree of the polynomial.

5. Now, that you have a conjecture. Let’s try to see if it always work. Let’s look at the graph of the polynomial .
1. Sketch the graph of the polynomial on the axes below. Remember that the shape of the graph is not as important as properly capturing how many times does the graph cross the x-axis.

2. Without needing the graph, what is the degree of the polynomial?

3. Now, going back to the graph, how many times does the graph actually cross the x-axis? (This is a bit of a trick question!)

4. Does this change your conjecture? Why?

5. Why do you think your conjecture might have changed? Can you figure out why? (You may not have an answer yet.)

6. Are you ready to change your conjecture yet? If so, to what?

6. Let’s try another polynomial. Let’s look at the graph of the polynomial .
1. What is the degree of this polynomial?

2. Now, let’s look at the graph of this polynomial. We had to change our window a little bit of the Graphing Calculator program. Here’s what a change in the window can show us about our graph.

3. Asking the trick question again, how many times does the graph cross the x-axis?

4. Has this graph changed your conjecture from Exercise #4? If so, how would you change it?

7. Let’s look at a different polynomial that might bring home the point a little more. Let’s look at the graph of the polynomial .
1. What is the degree of the polynomial?

2. Sketch the graph of the polynomial below. Remember to pay attention more to where the graph crosses the x-axis more than anything.

3. How many times does the graph cross the x-axis this time?

4. Do you have guesses why this might be the case? (Hint: have you tried to factor the polynomial yet?!)

5. Have you solidified your conjecture now?

8. Let’s try one more graph. Look at the graph of the polynomial . What happens to the graph of this function?

9. A final exercise.
1. Try to come up with a fourth-degree polynomial that crosses the x-axis five times.

2. Were you successful?

3. Does your conjecture hold up after seeing this exercise?