Download 2 - Kent

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?
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
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
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!
Let’s look at another
The standard form of the equation is:
The roots are:
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
2
3
4
The roots therefore are: -1, 2, 3
2 3 4
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!
This leads us to the
Rational Root Theorem
1. For polynomial
x 3  x 2  3x  3  0
Here p = -3 and q = 1
Factors of -3
±3, ±1

Or 3,-3, 1, -1
Factors of 1
±1
Possible roots are ___________________________________
3
2
2. For polynomial 3 x  9 x  4 x  12  0
Here p = 12 and q = 3
Factors of 12
±12, ±6 , ±3 , ± 2 , ±1 ±4

Factors of 3
±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???
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
 12
 12
1
6
 6
1
3
 3
1
2
 2
1
1
 1
1
4
 4
1
 12
 4
3
6
 2
3
3
 1
3
2
2

3
3
1
1

3
3
4
4

3
3
That is where our 9 possible answers come from!
Let’s Try One
Find the POSSIBLE roots of
5x3 - 24x2 + 41x - 20 = 0
Let’s Try One
5x3-24x2+41x-20=0
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.
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
Step 3 – factors
Factors of -3 = ±3, ±1
Factors of 1 = ± 1
Step 4 – possible
roots
-3, 3, 1, and -1
Step 5 – Test each
root
X
-3
3
Step 6 – synthetic
division
X³ + X² – 3x – 3
(-3)³ + (-3)² – 3(-3) – 3 = -12
-1
1
1
-3
-3
-1
0
3
0
-3
0
(3)³ + (3)² – 3(3) – 3 = 24
1
(1)³ + (1)² – 3(1) – 3 = -4
-1
(-1)³ + (-1)² – 3(-1) – 3 = 0
1
THIS IS YOUR ROOT
BECAUSE WE ARE LOOKING
FOR WHAT ROOTS WILL
MAKE THE EQUATION =0
1x² + 0x
-3
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
• 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
• Step 3 – factors
• Step 4 – possible roots
• Factors of -6 = ±1, ±2, ±3,
±6
Factors of 1 = ±1
• -6, 6, -3, 3, -2, 2, 1,
and -1
• Step 5 – Test each root
X
• Step 6 – synthetic
division
x³ – 5x² + 8x – 6
-6
-450
6
78
3
0 THIS IS YOUR ROOT
-3
-102
2
-2
-2
-50
1
-2
-1
-20
3
1
1
-5
8
-6
3
-6
6
-2
2
0
1x² + -2x + 2
• Step 7 – Rewrite
• x³ – 5x² + 8x – 6
= (x - 3)(x² – 2x + 2)
• Step 8– factor more
and solve
• (x - 3)(x² – 2x + 2)
X= 3
Quadratic Formula
x  1 i
• Roots are 3, 1 ± i
Irrational Root Theorem
•
•
•
•
Y  a0 x n  a1x n1  ...  an1x  an
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.
Complex pairs of form a + √ b and a - √ b
CONJUGATE ___________________________
1. For polynomial has roots 3 + √2
2
3 - √2 Degree of Polynomial ______
Other roots ______
2. For polynomial has roots -1, 0, - √3, 1 + √5
6
√3 , 1 - √5 Degree of Polynomial ______
Other roots __________
1. For polynomial has roots 1 + √3 and -√11
1 - √3
Other roots ______
√11
_______
4
Degree of Polynomial ______
Question: One of the roots of a polynomial is
Can you be certain that
4 2
4 2
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.
4 2
Write a polynomial given the roots
5 and √2
• Another root is - √2
• Put in factored form
• y = (x – 5)(x + √2 )(x – √2 )
Homework
• P. 316 12,14,18,19,21,25,28,33,34,38
Descartes' Rule of Signs
5
Decide what to FOIL first
y = (x – 5)(x + √2 )(x – √2 )
X
x
√2
X2
X √2
-√2
-X √2
-2
(x² – 2)
FOIL or BOX to finish it
up (x-5)(x² – 2)
y = x³ – 2x – 5x² + 10
Standard Form
y = x³ – 5x² – 2x + 10
x2
x
-5
-2
X3
-2x
-5x2
10
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
y = (x – √5 )(x + √5)(x – √7)(x + √7)
Foil or use a box method to multiply
the binomials
X
x
√5
X2
X √5
-√5
-X √5
-5
(x² – 5)
X
x
√7
X2
X √7
-√7
-X √7
-7
(x² – 7)
y = (x² – 5)(x² – 7)
x2
x2
FOIL or BOX to finish it
up
y = x4 – 7x² – 5x² + 35
Clean up
y = x4 – 12x² + 35
-7
-5
X4
-5x2
-7x2
35