1. Factor Theorem & Rational Root Theorem
Objective:
SWBAT find zeros of a polynomial by using Rational Root Theorem
(also known as Rational Zeros Theorem)

2. The Factor Theorem:
For a polynomial P(x), x – k is a factor iff P(k) = 0
iff
“if and only if”
It means that a theorem and its converse are true

3. If P(x) = x3 – 5x2 + 2x + 8, determine whether x – 4 is a factor.
remainder is 0, therefore yes
other factor

4. Terminology: Solutions (or roots) of polynomial equations
Zeros of polynomial functions
“k is a zero of the function f if f(k) = 0”
zeros of functions are the x values of the points
where the graph of the function crosses the x-axis
(x-intercepts where y = 0)

5. Ex 1: A polynomial function and one of its zeros are given, find the remaining zeros:

6. Ex 2: A polynomial function and one of its zeros are given, find the remaining zeros:

7. Rational Root Theorem:
Suppose that a polynomial equation with integral coefficients has the root p/q , where p and q are relatively prime integers. Then p must be a factor of the constant term of the polynomial and q must be a factor of the coefficient of the highest degree term.
(useful when solving higher degree polynomial equations)

8. Solve using the Rational Root Theorem:
4x2 + 3x – 1 = 0 (any rational root must have a numerator
that is a factor of -1 and a denominator
that is a factor of 4)
factors of -1: ±1
factors of 4: ±1,2,4
possible rational roots: (now use synthetic division
to find rational roots)
(note: not all possible rational roots are zeros!)

9. Listing Possible Rational Roots
When remembering how to find the list of all possible rational roots of a polynomial, remember the silly snake puts his tail over his head (factors of the “tail of the polynomial” over factors of the “head of the polynomial”).

10. Practice! This is how we LEARN…

11. Ex 3: Solve using the Rational Root Theorem:
possible rational roots:

12. Ex 4: Solve using the Rational Root Theorem:
possible rational roots:

13. Ex 5: Solve using the Rational Root Theorem:
possible rational roots:
To find other roots can use synthetic division
using other possible roots on these coefficients.
(or factor and solve the quadratic equation)

14. Section 3.3 – Polynomial Functions
Definition:
Multiplicity
The number of times a factor (m) of a function is repeated is referred to its multiplicity (zero multiplicity of m).
Zero Multiplicity of an Even Number
The graph of the function touches the x-axis but does not cross it.
Zero Multiplicity of an Odd Number
The graph of the function crosses the x-axis.

15. Section 3.3 – Polynomial Functions
Identify the zeros and their multiplicity
3 is a zero with a multiplicity of 1
Graph crosses the x-axis.
-2 is a zero with a multiplicity of 3
Graph crosses the x-axis.
-4 is a zero with a multiplicity of 1
Graph crosses the x-axis.
7 is a zero with a multiplicity of 2
Graph touches the x-axis.
-1 is a zero with a multiplicity of 1
Graph crosses the x-axis.
4 is a zero with a multiplicity of 1
Graph crosses the x-axis.
2 is a zero with a multiplicity of 2
Graph touches the x-axis.

16. Homework
3.3 #1-5 all, EOE, and 32