1. Factor Theorem & Rational Root Theorem
Objective:
SWBAT find zeros of a polynomial by using Rational Root Theorem

2. The Factor Theorem:
For a polynomial P(x), x – a is a factor iff P(a) = 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
“r is a zero of the function f if f(r) = 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. Ex 3: Solve using the Rational Root Theorem:
possible rational roots:

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

11. 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)