1. Entry Task I am greater than my square.
The sum of my numerator and denominator is 5
My numerator is a factor of 6
My denominator is a factor of 4
What fraction am I?
How did you find me?

2. 5.5 Theorems About Roots of Polynomial Equations
Target: I can solve equations using the Rational Root Theorem
I can use the Conjugate Root Theorem

3. Pick who will win the Lottery
Can you do it?
What if we could narrow it down?
What if I said it was someone in the room?

4. Let’s explore the following . . .
(x + 2)(x - 3)(x - 4) = 0
The standard form of the equation is:
x3 – 5x2 – 2x + 24 = 0
The roots are: -2, 3, 4

5. Take a closer look at the original equation and our roots:
x3 – 5x2 – 2x + 24 = 0
The roots are: -2, 3, 4
What do you notice?
-2, 3, and 4 all go into the last term, 24!

6. Let’s look at another The standard form of the equation is:
The roots are:

7. Take a closer look at the original equation and our roots:
24x3 – 22x2 – 5x + 6 = 0
This equation factors to:
(x + 1)(x - 2)(x - 3)= 0
The roots therefore are: -1, 2, 3
What do you notice?
The numerators 1, 2, and 3 all go into the last term, 6!
The denominators (2, 3, and 4) all go into the first term, 24!

8. This leads us to the Rational Root Theorem

9. Example ±3, ±1 ±1 ±12, ±6 , ±3 , ± 2 , ±1 ±4 ±1 , ±3 1. For polynomial
Here p = -3 and q = 1
Factors of -3
Factors of 1
±3, ±1 ±1 
Or 3,-3, 1, -1
Possible roots are ___________________________________
2. For polynomial
Here p = 12 and q = 3
Factors of 12
Factors of 3
±12, ±6 , ±3 , ± 2 , ±1 ±4
±1 , ±3 
Possible roots are ______________________________________________
Or ±12, ±4, ±6, ±2, ±3, ±1, ± 2/3, ±1/3, ±4/3
Wait a second Where did all of these come from???

10. Let’s look at our solutions
±12, ±6 , ±3 , ± 2 , ±1, ±4
±1 , ±3
Note that + 2 is listed twice; we only consider it as one answer
Note that + 1 is listed twice; we only consider it as one answer
Note that + 4 is listed twice; we only consider it as one answer
That is where our 9 possible answers come from!

11. Find the POSSIBLE roots of
Let’s Try One
Find the POSSIBLE roots of
5x3 - 24x2 + 41x - 20 = 0

12. Let’s Try One
5x3-24x2+41x-20=0

13. That’s a lot of answers!
Obviously 5x3 - 24x2 + 41x – 20 = 0 does not have all of those roots as answers.
Remember: these are only POSSIBLE roots. We take these roots and figure out what answers actually WORK.

14. Step 1 – find p and q
p = -3
q = 1
Step 2 – by RRT, the only rational root is of the form…
Factors of p
Factors of q

15. Step 3 – factors
Factors of -3 = ±3, ±1
Factors of 1 = ± 1
Step 4 – possible roots
-3, 3, 1, and -1

16. Step 6 – synthetic division
Step 5 – Test each root
Step 6 – synthetic division
X X³ + X² – 3x – 3 -1 -3 3 1 -1
(-3)³ + (-3)² – 3(-3) – 3 = -12
(3)³ + (3)² – 3(3) – 3 = 24 3 -1
(1)³ + (1)² – 3(1) – 3 = -4 1 -3
(-1)³ + (-1)² – 3(-1) – 3 = 0
THIS IS YOUR ROOT
BECAUSE WE ARE LOOKING
FOR WHAT ROOTS WILL
MAKE THE EQUATION =0
1x² + 0x -3

17. Step 7 – Rewrite
x³ + x² - 3x - 3
= (x + 1)(x² – 3)
Step 8– factor more and solve
(x + 1)(x² – 3)
(x + 1)(x – √3)(x + √3)
Roots are -1, ± √3

18. Using the Polynomial Theorems FACTOR and SOLVE x³ – 5x² + 8x – 6 = 0
Step 1 – find p and q
p = -6
q = 1
Step 2 – by RRT, the only rational root is of the form…
Factors of p
Factors of q

19. Using the Polynomial Theorems FACTOR and SOLVE x³ – 5x² + 8x – 6 = 0
Step 3 – factors
Factors of -6 = ±1, ±2, ±3, ±6
Factors of 1 = ±1
Step 4 – possible roots
-6, 6, -3, 3, -2, 2, 1, and -1

20. Using the Polynomial Theorems FACTOR and SOLVE x³ – 5x² + 8x – 6 = 0
Step 6 – synthetic division
Step 5 – Test each root
X x³ – 5x² + 8x – 6 -6 6 3 -3 2 -2 1 -1
-450 78
-102 -2
-50
-20 3
THIS IS YOUR ROOT 6 3 -6 1 -2 2
1x² + -2x + 2

21. Using the Polynomial Theorems FACTOR and SOLVE x³ – 5x² + 8x – 6 = 0
Step 7 – Rewrite
x³ – 5x² + 8x – 6
= (x - 3)(x² – 2x + 2)
Step 8– factor more and solve
(x - 3)(x² – 2x + 2)
Roots are 3, 1 ± i
X= 3
Quadratic Formula

22. Irrational Root Theorem
For a polynomial
If a + √b is a root,
Then a - √b is also a root
Irrationals always come in pairs. Real values do not.
CONJUGATE ___________________________
Complex pairs of form a + √ b and a - √ b

23. Example (IRT) 1. For polynomial has roots 3 + √2 2 3 - √2
Other roots ______
Degree of Polynomial ______
2. For polynomial has roots -1, 0, - √3, 1 + √5 6
√3 , 1 - √5
Other roots __________
Degree of Polynomial ______

24. Example (IRT) 1. For polynomial has roots 1 + √3 and -√11 1 - √3 √11
Other roots ______ _______ 4
Degree of Polynomial ______
Question: One of the roots of a polynomial is
Can you be certain that is also a root?
No. The Irrational Root Theorem does not apply unless you know that all the coefficients of a polynomial are rational. You would have to have
as your root to make use of the IRT.

25. Write a polynomial in standard form with the given the roots 5 and √2
Another root is - √2
Put in factored form
y = (x – 5)(x + √2 )(x – √2 )
y = x3 – 5x2 – 2x +10

26. Homework
Rational Roots Wkst

27. Descartes' Rule of Signs5

29. Decide what to FOIL first y = (x – 5)(x + √2 )(x – √2 )-2

30. FOIL or BOX to finish it up (x-5)(x² – 2)
y = x³ – 2x – 5x² + 10
Standard Form
y = x³ – 5x² – 2x + 10 x x -5 X3
-2x
-5x2 10

31. Write a polynomial given the roots -√5, √7
Other roots are √5 and -√7
Put in factored form
y = (x – √5 )(x + √5)(x – √7)(x + √7)
Decide what to FOIL first

32. y = (x – √5 )(x + √5)(x – √7)(x + √7)
Foil or use a box method to multiply the binomials
X √5
X √7 x √5 x √7 X2
-X √5 X2
-X √7
X √5 -5
X √7 -7
(x² – 5)
(x² – 7)

33. FOIL or BOX to finish it up y = x4 – 7x² – 5x² + 35 Clean up -7 X4
-5x2
-7x2 35