1. Factor and Rational Roots
Theorems

2. Factor Theorem
A Polynomial P(x)
-only has a factor (x - a) if the value of P(a) is 0 (no remainder)
Example
3
a) Determine whether x + 2 is a factor of f(x) = x - 6x - 4
b) Determine the other factors of f(x)

3. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Procedure Example
Step 1: Find all possible ƒ(x) = 3x3 - 4x2 - 5x + 2
numerators by listing the
positive and negative 1, -1, 2, -2
factors of the constant
term.

4. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Procedure Example
ƒ(x) = 3x3 - 4x2 - 5x + 2
Step 2: Find all possible
denominators by listing
the positive factors of the 1, 3
leading coefficient.

5. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Procedure Example
Step 3: List all possible ƒ(x) = 3x3 - 4x2 - 5x + 2
rational roots. Eliminate
all duplicates. 1, -1, 2, -2
1, 3

6. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Procedure Example
ƒ(x) = 3x3 - 4x2 - 5x + 2
Step 4: Use synthetic division and
the factor theorem to reduce ƒ(x)
to a quadratic. (In our example,
weʼll only need one such root.)
- 1 is a root!
So,

7. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Procedure Example
Step 5: Factor the quadratic.
Step 6: Find all roots.

8. Step 1: Find all possible Step 4: Use synthetic division and
numerators by listing the the factor theorem to reduce ƒ(x)
positive and negative to a quadratic. (In our example,
factors of the constant weʼll only need one such root.)
term.
Step 2: Find all possible Step 5: Factor the quadratic.
denominators by listing
the positive factors of the
leading coefficient.
Step 6: Find all roots.
Step 3: List all possible
rational roots. Eliminate
all duplicates.
You try!!! Find all of the factors and roots of this
polynomial
ƒ(x) = x3 + 3x 2 - 13x - 15