1.

Warm – up #4 Synthetic Div

x = 𝑥3 𝑥2 𝑥¿−1 −2 −3 0

−2

2

0

0

−33

3𝑥2 𝑥¿ 𝑅

𝑥4

1

1𝑥3

−1

=

Homework Log

Wed

12/2

Lesson 5 – 5

Learning Objective: To apply Rational Root Thm (& Descartes’Rule)

Hw: W.S. 5-5

12/2/15 Lesson 5 – 5 Rational Root Theorem

Day 1Algebra II

To understand & apply the Rational Root Theorem

To understand & apply the Conjugate Root Theorem

(To understand and apply Descartes’ Rule of Signs)

Learning Objective

List all possible rational zeros++…+

Rational Root Theorem

q = factors of p = factors of

Possible Rational Zeros = combos

Look at COEFFICIENTS

1. List all possible rational zeros

Rational Root Theorem

q = 1 p = 12

=

= 2, 3, 4, 6, 12

2. List all possible rational zeros

Rational Root Theorem

q = 3 p = 10

=

= 2, , , , ,

Divide each numerator by each denominator

112151

101

1323

53

103

3. What are the rational roots of

Applying Rational Root Theorem

q = 1 p = 42

Step 1: List all the possible rational zeros

=

= 2, , , , ,

3. What are the rational roots of

Applying Rational Root Theorem

2, , , , ,

Step 2: Test each possible rational root in original function until you find a root.

Do synthetic substitution until get R0

3.

Applying Rational Root Theorem

1𝑥3 𝑥2 𝑥 ¿1 −6 −13 42

11−5−5−18

−1824𝑅

-11 −6 −13 42

1−1−77−6648𝑅

Nope! 1 is NOT a root!

Nope! 1 is NOT a root!

Try something else!

3.

Applying Rational Root Theorem

2𝑥3 𝑥2 𝑥 ¿1 −6 −13 42

12−4−8−21

−420 YES! 2 IS a

root!

Factor & Solve to find rest of the roots! (x-2)(x – 7)(x + 3)

x = 2, 7, –3

𝑥2 𝑥¿ 𝑅

4. What are the rational roots of

Applying Rational Root Theorem

q = 2 p = 6

=

= 2, , , ,

4.

Applying Rational Root Theorem

-1𝑥3 𝑥2 𝑥 ¿21 −7 −6

2−2−11−660 YES! -1 IS a root!

We have (x+1)( Factor & Solve to find rest of the roots! (x+1)(2x + 3)(x - 2)

x = -1,, 2

𝑥2 𝑥¿ 𝑅

If is a root of P(x), then is also a root of P(x)

If is a root of P(x), then is also a root of P(x)

Conjugate Root Theorem

5. and are roots of P(x), find two other roots

and

6. and are roots of P(x), find one other root.

Conjugate Root Theorem

Stop here! Do rest tomorrow!

The biggest exponent of tells you how many roots there are

The # of times changes signs tells you how many possible (+) roots there are or less that by an even #

The # of times changes signs tells you how many possible (-) roots there are or less that by an even #

Descartes’ Rule of Signs

7.

Descartes’ Rule of Signs

2 Sign Change

1 Sign Change+ - 2 1 00 1 2

, so should have 3 solutions total

Sum = 3

Sum = 3

8.

Descartes’ Rule of Signs

5 Sign Change

4 Sign Change

8. , so should have 9 solutions totalDescartes’ Rule of Signs

+ -

5 4 05 2 25 0 43 4 23 2 43 0 61 4 41 2 61 0 8

It could have 5 positive roots, 4 negative roots, & 0 imaginary roots

It could have 3 positive roots, 2 negative roots, & 4 imaginary roots

9.

Descartes’ Rule of Signs

3 Sign Change

1 Sign Change+ - 3 1 01 1 2

It could have 3 + roots, 1 - root, & 0 imaginary roots

It could have 1 + roots, 1 - root, & 2 imaginary roots

10. What are the rational roots of

Applying Rational Root Theorem

q = 1 p = 6

=

= 2, ,

10.

Applying Rational Root Theorem

1𝑥3 𝑥2 𝑥 ¿1 −1 6 −6

1100660 YES! 1 IS a root!

is not factorable, so the only RATIONAL ROOT is 1

*Can solve = 0 to find irrational roots

𝑥2 𝑥¿ 𝑅

x =1,

What does Descartes’ Rule of Signs say about the # of positive, negative, & imaginary roots of

Ticket Out the Door

Assignment:

W.S. 5-5