1. 2.6 Find Rational Zeros pg. 128 What is the rational zero theorem?
What information does it give you?

2. The rational zero theorem…
If f(x)=anx a1x+a0 has integer coefficients, then every rational zero of f has the following form:
p factor of constant term a0
q factor of leading coefficient an n =

3. Example 1: Find rational zeros of f(x)=x3+2x2-11x-12 List possible
LC= CT=-12
X= ±1/1,± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1
Test:
X= x=
Since -1 is a zero: (x+1)(x2+x-12)=f(x)
Factor: (x+1)(x-3)(x+4)=0
x= x=3 x=-4

4. List the possible rational zeros of f using the rational zero theorem.
a. f (x) = x3 + 2x2 – 11x + 12
Factors of the constant term: ± 1, ± 2, ± 3, ± 4, ± 6, ± 12
Factors of the leading coefficient: ± 1
Simplified list of possible zeros: ± 1, ± 2, ± 3, ± 4, ± 6, ± 12

5. Factors of the constant term: + 1, + 2, + 5, + 10
f (x) = 4x4 – x3 – 3x2 + 9x – 10
Factors of the constant term: + 1, + 2, + 5, + 10
Factors of the leading coefficient: + 1, + 2, + 4

6. Extra Example: Find rational zeros of: f(x)=x3-4x2-11x+30 LC=1 CT=30
Test:
x= x=
X= (x-2)(x2-2x-15)=0
(x-2)(x+3)(x-5)=0
x= x= x=5

7. Find all real zeros of f (x) = x3 – 8x2 +11x + 20.
SOLUTION
STEP 1

8. STEP 2
Test these zeros using synthetic division. –
Test x =1:
1 – –
1 is not a zero.
Test x = –1:
– – – –
–1 is a zero

9. Because –1 is a zero of f, you can write f (x) = (x + 1)(x2 – 9x + 20).
STEP 3
Factor the trinomial in f (x) and use the factor theorem.
f (x) = (x + 1) (x2 – 9x + 20)
= (x + 1)(x – 4)(x – 5)
The zeros of f are –1, 4, and 5.
ANSWER

10. Assignment 2.6
p. 132, 1, 4-22 even, 36-37

11. Find Zeros -leading coefficient is not 1
f(x)=10x4-3x3-29x2+5x+12
List: LC= CT=12
x= ± 1/1, ± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1, ± 3/2, ± 1/5, ± 2/5, ± 3/5, ± 6/5, ± 12/5, ± 1/10, ± 3/10, ± 12/10
w/ so many –sketch graph on calculator and find reasonable solutions:
x= -3/2, -3/5, 4/5, 3/2
Check:
x= -3/
Yes it works
* (x+3/2)(10x3-18x2-2x+8)*
(x+3/2)(2)(5x3-9x2-x+4) -factor out GCF
(2x+3)(5x3-9x2-x+4) multiply 1st factor by 2

12. Repeat finding zeros for:

13. If the highest degree is more than 3
If the highest degree is more than 3 (like 4) you will need to do synthetic division again, this time on the “new” equation you just found. Your goal is to divide your equation down to a 2nd degree equation so you can factor or use the quadratic formula. Each time you do synthetic division, your equation goes down 1 degree.

14. What is the rational zero theorem?
If f(x)=anx a1x+a0 has integer coefficients, then every rational zero of f has the following form:
p factor of constant term a0
q factor of leading coefficient an
What information does it give you?
It gives you a pool of numbers to use to help you find a divisor. =

15. Assignment 2.6
p. 132, even, all