1. FINDING ROOTS WITHOUT A CLUE
Rational Zero Theorem: If
has integer coefficients, then every rational zero of f has the following form:
p/q=factors of constant term/factors of leading coefficient term

2. USING THE RATIONAL ZERO THEOREM
List the possible rational zeros of the function using the rational zero theorem: 1. 2. 3. 4.

3. Find all of the roots of: g(x) = x4 + 2x3 – 5x2 – 4x + 6
(-3 and 1 using the Rational Zero Theorem )
Find all of the roots of:
g(x) = x4 + 2x3 – 5x2 – 4x + 6
ANSWER: -3, 1, ,

4. Find all of the roots of: g(x) = 4x3 – 16x2 + 11x + 3
(3 using the Rational Zero Theorem)
Find all of the roots of:
g(x) = 4x3 – 16x2 + 11x + 3
ANSWER: 3, ,

5. Find all of the roots of: g(x) = x3 – x2 – 11x + 3
(-3 using the Rational Root Theorem)
Find all of the roots of:
g(x) = x3 – x2 – 11x + 3
ANSWER: -3, ,

6. Conjugate Pairs
If a + bi is a zero
Then a – bi is a zero

7. Imaginary zeros don’t cross x-axis
Imaginary zeros don’t cross x-axis!!!! They are imaginary you can’t see them on a graph.

8. Find all of the zeros. zeros: x = 0 x = 2i x = -2i
Always look to see if a function will factor.
This function WILL factor.
zeros: x = x = 2i x = -2i

9. x = -1, 2 from the Rational Root Theorem
Find all the roots
x = -1, 2 from the Rational Root Theorem
f(x) = x4 – 5x3 + 7x2 + 3x - 10 -1 1 -5 7 3
-10 -1 6
-13 10 2 1 -6 13
-10 2 -8 10 1 -4 5
These are all the roots

10. Find all of the zeros of the function:
5 roots:
3 real
2 imag.
BOUNCES!
x = -2
x = 1
x = 1

11. x = -2
x = 1 -2 1 1 2
-12 8
x = 1 -2 4
-10 16 -8 1 1 -2 5 -8 4 1 -1 4 -4
Zeros:: 1 1 -1 4 -4
x = -2
x = 1 1 4
x = 1
x = 2i x = -2i 1 4
f(x) = x2 + 4
0 = x2 + 4
-4 = x2
-2i, 2i = x