Use a root’s multiplicity to describe the polynomial graph’s behavior at an x-intercept.
Understand that a polynomial of degree n has exactly n complex zeros and can be written as a product of n linear factors.
Find all zeros of a polynomial function when given in factored form; identify when zeros will be imaginary based on the polynomial's graph or equation in factored form.
Quick Lesson Plan
In today’s activity, students look at three polynomial functions and their graphs and notice patterns between the multiplicity of a zero and the behavior of the graph at that zero. They also articulate the connection between a polynomial in factored form and the x-intercepts of that polynomial. Note that while many of these topics are covered in an Algebra 2 class, this lesson combines ideas from several lessons in Algebra 2. For example, in questions 3-5, students are already looking at polynomials with complex zeros that do not appear on the graph and relating the number of zeros to the degree of a polynomial.
What do you mean when you say the x-intercepts are the opposite of the factors?
How are the solutions to the equation f(x)=0 related to the x-intercepts?
What do you think the graph would look like if the factor (x-2) was raised to the 5th instead of the 3rd power?
How can you tell from a graph whether there will be imaginary solutions or only real solutions?
The first four lessons of this unit focus on a polynomial’s function rate of change (Lesson 2.1), zeros (2.2), symmetry (2.3) and end behavior (2.4). In this lesson we review key concepts from Algebra 2 about a function’s zeros and their multiplicity, but also address the number of zeros using the Fundamental Theorem of Algebra. Note that this lesson is NOT about finding all zeros from standard form. AP Precalculus has a unique sequence in that it saves more complex function manipulation and processes like polynomial division until the very end of the unit, rather than including it in the topics where it is most often applied (like finding all zeros of a polynomial). The goal is to focus on conceptual understanding of the number of zeros, and the type of zeros (real or imaginary), and how these are linked to a function’s graph. For this reason, students will only be given polynomials that are already in factored form in this lesson. The factors will be either linear or quadratic.
A crucial piece of today’s lesson is that students understand the relationship between the factors of a polynomial function and its zeros as well as the zeros of a polynomial function and its x-intercepts. It is not sufficient for students to say that the x-intercepts are the “opposite” of the factors! The x-intercepts are the values that make each factor equal to zero. The zero product property says that if one factor is 0, then the product is zero. If a polynomial function has an output of 0, then the y-value is 0, which means the function is passing through the x-axis. This is the line of reasoning that we expect students to be able to articulate at the AP Precalculus level. If you find that your students are far from this goal, we strongly recommend using this Algebra 2 lesson about the connection between factors and zeros before you do Lesson 2.2.
Finally, while x-intercepts and zeros are often used interchangeably (and do have a lot of overlap) there are some distinctions. All x-intercepts are zeros (input values that make the output equal to 0) but not all zeros are x-intercepts (complex zeros are solutions to f(x)=0 but they are not visible on a graph as an x-intercept). In short, x-intercepts are real zeros.