Download 2.4 Zeros of Polynomial Functions Obj: To use the Rational Root

2.4
Zeros of Polynomial Functions
Obj: To use the Rational Root Theorem to identify the possible rational roots, and determine the number of
positive and negative roots a polynomial has
DOL: I will correctly solve 1 problem
Do Now: If you have 2 pants and 3 shirts to choose an outfit from, how would you find the number of possible
combinations could you make? What is it?
Consider the following polynomial
𝑃(𝑥) = 2𝑥 3 − 𝑥 2 − 25 ….. If 𝑃(𝑥) = 0:
- How many roots (complex roots)?
- Of the roots, how many are rational? Can we figure this out using what we have already learned?
𝑝
Translation: If a Rational Root exists, it can be obtained by using , where “𝑝” is the factors of the constant
𝑞
term, and “𝑞” is the factors of the Leading Coefficient.
Example 1: Given
𝑃(𝑥) = 2𝑥 3 − 𝑥 2 − 25,
find all POSSIBLE roots.
𝑝  25:
𝑞  2:
Now find ALL solutions of the equations. If any rational roots exist…it would be one (or more) of the above.
Once the polynomial
is depressed to a
quadratic…we have
several ways to find
the rest of the roots.
Descartes Rule of Signs – used to determine the possible numbers and combinations of positive and negative
real zeros, by counting the sign changes of f(x) and f(-x)
Steps: (arrange powers in descending order, no place holders for missing powers, what is the total number of
roots)
1.)
# of POSITIVE REAL ROOTS = the # of sign changes in f(x)
(or less by an even number: 2, 4, …)
2.)
# of NEGATIVE REAL ROOTS = the # of sign changes in f(-x)
(or less by an even number: 2, 4, …)
3.)
Make a chart to figure out how many IMAGINARY ROOTS (always come in pairs)
Try for the previous equation: 𝑃(𝑥) = 2𝑥 3 − 𝑥 2 − 25,
P:
N:
Ex 2: Find the possible positive, negative, and imaginary zeros for roots:
𝑓(𝑥) = 𝑥 4 − 2𝑥 3 + 7𝑥 2 + 4𝑥 − 15
P
N
I
Ex 3: For the function below:
a) Determine the number of possible positive and negative real zeros (make a chart)
𝑝
b) List all possible rational zeros (use 𝑞)
c) Given one of the zeros/roots, find the remaining zeros/roots
𝑓(𝑥) = 𝑥 4 + 4𝑥 3 − 12𝑥 − 9 𝐺𝑖𝑣𝑒𝑛: −1 𝑎𝑛𝑑 − 3 𝑎𝑟𝑒 𝑧𝑒𝑟𝑜𝑠
Practice:
1) For the function below:
a) Determine the number of possible positive and negative real zeros (make a chart)
𝑝
b) List all possible rational zeros (use 𝑞)
c) Find all zeros – remember what your chart told you about your answers!!
𝑥 3 + 8𝑥 2 + 16𝑥 + 5 = 0