Algebra 2 Note-taking Guide
Algebra 2 - Lesson 4.04 Rational Root Theorem and Descartes’ Rule of Signs
Please print this out in advance, and as you are working through the lesson, fill in the information and use this as your notes.
þ The goal is to have all the empty boxes checked
As you complete this lesson, please check that you can answer:
¨ What do the Rational Root Theorem and Descartes’ Rule of Signs indicate about the zeros of a polynomial function?
¨ How can the zeros and end behavior of a polynomial function allow a graph to be sketched?
The Rational Root Theorem (page 1)
Take a look at the function
The Fundamental Theorem of Algebra says the degree of this function is equivalent to the number of zeros the function has. There are ______zeros.
The Remainder Theorem says a possible zero can be tested using synthetic division.
If the tested number creates a remainder of zero in the synthetic division process, then the factor theorem states that the tested number is a ______of the function.
Use the space below to determine if 1 is a zero of the function by testing it with synthetic division:
The zeros of the function are found by setting the factors3x−1and2x+7equal to zero and solving forx.
f(x) / = / (3x−1)(2x+7)0 / = / (3x−1)(2x+7)
Finish the process below:
Notice that the numerators of these solutions (1 and −7) are factors of the constant term (−7). The denominators of these solutions (3 and 2) are factors of the leading coefficient (6).
TheRational Root Theoremstates that a list of potential rational zeros of a polynomial function is found by listing the factors of the constant term, defined as_____, and dividing those factors by each of the factors of the leading coefficient, defined as______.
To find all of the possible rational zeros of the polynomial function, divide each factor of p by each factor of q. In other words, find all possible combinations of pq.
pq=______=
Keep in mind that ±1, ±2, ±3, and ±6 are only thepossible ______zeros. Irrational and complex zeros cannot be identified by thepqvalues.
Also, not all of these eight possible zeros will actually be zeros. Remember, the Fundamental Theorem of Algebra tells us that the function has exactly ______zeros.
To find the zeros of the function, test each of the possible zeros in the synthetic division process to discover which results in a remainder of ______. Use the space below to show your work.
Of the eight possible rational zeros, only three are actual zeros of the polynomial function: ____, ____ and ____.
Step 1: Find the factors of p.Step 2: Find the factors of q.
Step 3: Write all factors of p over q.
Step 4: Simplify.
Be sure to complete the two additional examples for extra practice:
Descartes' Rule of Signs (page 2)
It was determined, by the Fundamental Theorem of Algebra, that there are five values of x that makeequal to zero. These values for x are also known as the zeros of the function. Using the Rational Root Theorem and synthetic division, 2, −1, and −3 were found to be three of the five zeros of the function. Notice that these three zeros, which are also known as roots, are thex-intercepts of the graph of this function.
But what are the other two zeros?
Think about the name "Rational Root Theorem." The term "rational" describes numbers that can be written as fractions. This means that ______numbers and______numberscannot be found using this theorem.
Complex zeros can only be found using the ______or by ______.
Using our sample function , fill in the table below with detailed steps for how to find the number of positive and negative real zeros.
Use the space below to practice with examples 1 and 2 (page 3):
Creating sketches of graphs from zeros (page 4)
Using what has been discovered about possible zeros, and combining it with the y-intercept, it is possible to create rough sketches of the polynomial functions.
Watch the video example at the top of page 4 and use the graph below to sketch the function as you follow along with the video.
Finding the y-intercept (page 4)
To find the y-intercept, substitute 0 for ____ and solve for ____.
What is the y-intercept of ?
Finding the zeros (page 4)
Using our same function , apply the theorems you learned:
Descartes’ Rule of Signs:Rational Root Theorem:
Uncovering the Zeros:
Use the grid below to sketch the graph of .
Activity 1 (page 5)Activity 2 (page 5)
Activity 3 (page 5)
Finally, complete the 4.04 Assessment, Rational Root Theorem and Descartes’ Rule of Signs. This is an auto-graded assignment. You will get immediate feedback on your work.
Algebra 2 Notetaking Guide
Version 14
Florida Virtual School