Curve Sketching
Sketch #6 by flickr user rbanks

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?

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.

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.

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

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,

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.

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

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