Factor and Rational Roots

Theorems

A Polynomial P(x)

-only has a factor (x - a) if the value of P(a) is 0 (no remainder)

Example

a) Determine whether x + 2 is a factor of f(x) = x3 - 6x - 4

b) Determine the other factors of f(x)

Factor Theorem

Rational Roots Theorem

if P(x) has rational roots, they may be found using this procedure:

Procedure Example

Step 1: Find all possible numerators by listing the positive and negative factors of the constant term.

For any polynomial function

ƒ(x) = 3x - 4x - 5x + 23 2

1, -1, 2, -2

Rational Roots Theorem

if P(x) has rational roots, they may be found using this procedure:

Procedure Example

For any polynomial function

ƒ(x) = 3x - 4x - 5x + 23 2Step 2: Find all possible denominators by listing the positive factors of the leading coefficient.

1, 3

Rational Roots Theorem

if P(x) has rational roots, they may be found using this procedure:

Procedure Example

For any polynomial function

ƒ(x) = 3x - 4x - 5x + 23 2Step 3: List all possible rational roots. Eliminate all duplicates. 1, -1, 2, -2

1, 3

Rational Roots Theorem

if P(x) has rational roots, they may be found using this procedure:

Procedure Example

For any polynomial function

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.)

So,

-1 is a root!

ƒ(x) = 3x - 4x - 5x + 23 2

Rational Roots Theorem

if P(x) has rational roots, they may be found using this procedure:

Procedure Example

For any polynomial function

Step 5: Factor the quadratic.

Step 6: Find all roots.

ƒ(x) = x + 3x - 13x - 153 2

You try!!! Find all of the factors and roots of this polynomial

Step 1: Find all possible numerators by listing the positive and negative factors of the constant term.

Step 2: Find all possible denominators by listing the positive factors of the leading coefficient.

Step 3: List all possible rational roots. Eliminate all duplicates.

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.)

Step 5: Factor the quadratic.

Step 6: Find all roots.