1. Finding Rational Zeros

2. Zeros = Solutions = Roots = x-intercepts
Find all zeros of x2 –10x + 24
You just factor and set each factor to zero.
(You knew this already!)
(x – 12)(x + 2) = 0
x = 12, -2
You could also graph and see where it crosses the
x-axis (x-intercepts)
(You knew this already too!)

3. Refresh. What is a Rational Number?
A number that can be written as the ratio of two integers.
Examples:
It can also be an ending or repeating decimal.
Examples: … …

4. = Rational Zero Theorem
If f(x) = anxn + … + a1x + a0 (it’s a polynomial)
and the polynomial has integer coefficients, then
EVERY rational zero of f has the following form: =
factor of the constant term .
factor of the leading coefficient

5. = p is all of the factors of the constant term. 1, 3, 2, 4, 6, 12
“p over q”
Find the rational zeros of f(x) = x3 + 2x2 – 11x – 12
p is all of the factors of the constant term. 1, 3, 2, 4, 6, 12
q is all of the factors of the leading coefficient.
This one is easy because the leading coefficient is 1 !
The only factor is: 1 = 1, 3, 2, 4, 6, 12

6. Using
Still finding the rational zeros of f(x) = x3 + 2x2 – 11x – 12 1, 3, 2, 4, 6, 12
Possible Zeros:
Do synthetic division until you find a zero.
x x (-11) x + (–12) – 1
k-value
1 •
1 is not a zero to this function.
Remainder is not zero so 1 3 -8 1 3 -8
-20

7. Keep trying Possible Zeros
Test x = -1.
x x (-11) x + (–12)
-1 • – -1
k-value
-1 IS a zero to this function!
Remainder IS zero so -1 -1 12 1 1
-12
Since -1 is a zero of f, then the result is a factor (x2 + x – 12)
This is factorable into: (x + 4)(x – 3).
The zeros are: -1 (original zero), -4, 3

8. The Nightmare Example
Find the rational zeros of f(x) = 10x4 - 3x3 - 29x2 + 5x + 12 = 1, 3, 2, 4, 6, 12 1 1, 3, 2, 4, 6, 12 2 1, 3, 2, 4, 6, 12 5 1, 3, 2, 4, 6, 12 10

9. The Nightmare Example (cont’d)
Finding the zeros of f(x) = 10x4 - 3x3 - 29x2 + 5x + 12
With so many possible zeros, it’s worth our time to get a ballpark answer by graphing the polynomial on the calculator.
-3/2 •
-15 27 3
-12 10
-18 -2 8
We found the 1st Zero!

10. = What do you need to remember? Rational Zero Theorem
If f(x) = anxn + … + a1x + a0 (it’s a polynomial)
and the polynomial has integer coefficients, then
EVERY rational zero of f has the following form:
factor of the constant term .
factor of the leading coefficient =
Rational Zero Theorem
Be able to list all possible rational zeros.
Then let your calculator do the rest!