1. Essential Question: How do I use synthetic division to find all of the zeros?
What is synthetic division?
How do I find all the factors given 1 root?

2. Use Synthetic Division to divide
x4 – 10x2 – 2x by (x + 3) -3 1
-10 -2 4 -3 9 3 -3 1 -3 -1 1 1
remainder

3. You Try!!

4. Factors vs. Roots
Factors: What multiplies together to get the polynomial. The things in ( )
ex., (x+2), (2x-3), (x-8)
Roots: What the x values equal. After setting the factor =0. Can also be called zeros, solutions, or x-intercepts.
ex., x=-2, x=3/2, x=8
The can also be written as: f(-2)=0, f(3/2)=0,f(8)=0

5. Factor Theorem
If remainder = 0 when we plug a #, root, into some function, then (x – #) is a factor.

6. Before…
We could get a remainder using long division or synthetic division
The last number using synthetic division is your remainder!
We want a remainder of 0 to find the other roots…and we want to be able to find other roots!

7. Chapters 1 & 2 The Conservative Tradition in Educational Thought
Determine whether (x + 2) is a factor of

8. You Try!! Determine whether (x + 4) is a factor of
f(x) = x3 + 8x2 + 8x - 32

9. Chapters 1 & 2 The Conservative Tradition in Educational Thought
If (x-1) is a factor, find all the factors and roots of g(x) = 3x3 + 8x2 – 3x – 8 1 3 8 -3 -8 3 11 8 3 11 8
Depressed Polynomial
3x2
+11x +8
FACTOR Using Big X
All FACTORS:
Roots: x=1,x=-1,x=-8/3

10. (x-2) is a factor (x+3) is a factor
Given that f(2)=0 and x=-3 are zeros of
Find all the factors and zeros 2 2 7 -4
-27
-18 4 22 36 18
(x-2) is a factor -3 2 11 18 9 -6
-15 -9
(x+3) is a factor 2 5 3
Zeros: f(2)=0, x=-3,x=-1, f(-3/2)=0

11. You Try!! Given that (x-10) is a factor of , find all the
Factors and roots.
Zeros: f(10)=0, x=4, x=-2

12. You Try
Given that f(2)=0 Find the factors and zeros of:

13. You try:
Given that x=1 is a zero of the function: Find all the factors and roots.