Pre-Cal 30S January 14, 2009

Rational Roots Theorem(really this time)

At the Feet of an Ancient Master by flickr user premasagar

Determine each value of k.(a) When x + kx + 2x - 3 is divided by x + 2, the remainder is 1.3 2

Determine each value of k.(a) When x + kx + 2x - 3 is divided by x + 2, the remainder is 1.3 2

Determine each value of k.(b) When x - kx + 2x + x + 4 is divided by x - 3, the remainder is 16.4 3 2

(b) What is the remainder when the polynomial is divided by x - 2?

(a) Determine the value of b.

When the polynomial 2x + bx - 5 is divided by x - 3, the remainder is 7.2

Rational Roots Theorem

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

Procedure ExampleStep 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, we’ll 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.

Rational Roots TheoremYou try ...

ƒ(x) = x + 3x - 13x - 153 2