The document describes the rational roots theorem and procedure for finding rational roots of a polynomial function. It involves: 1) Listing all possible numerator and denominator factors of the constant term and leading coefficient. 2) Writing all possible rational roots. 3) Using synthetic division and the factor theorem to reduce the polynomial to a quadratic. 4) Factoring the quadratic to find rational roots.

1. Curve Sketching
Sketch #6 by flickr user rbanks

2. When the polynomial 2x 2 + bx - 5 is divided by x - 3, the remainder is 7.
(a) Determine the value of b.
(b) What is the remainder when the polynomial is divided by x - 2?

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

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

6. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Example
Procedure
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,
we’ll only need one such root.)
is a root!
-1
So,

7. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Example
Procedure
ƒ(x) = 3x3 - 4x2 - 5x + 2
Step 5: Factor the quadratic.
Step 6: Find all roots.

8. Rational Roots Theorem
You try ...
ƒ(x) = x3 + 3x 2 - 13x - 15

9. Rational Roots Theorem
You try ... ƒ(x) = x3 - 17x + 4