1. Work on example #1 in today’s packet.
pencil, red pen, highlighter, notebook
M3D15
Have out:
Bellwork:
Work on example #1 in today’s packet.

2. Introduction to Synthetic Division
Recall from yesterday that we used long division when dividing polynomials.
Example #1: Let P(x) = 2x3 + 3x2 – 15x – 16 and d(x) = x – 3,
find – + – + – +

3. coefficients of quotient q(x)
P(x) = 2x3 + 3x2 – 15x – 16 d(x) = x – 3
There is a nice shortcut to long division called ________________. Synthetic division is for divisors of the form _____. Let’s try the problem using this new method.
synthetic division
x – k
coefficients of dividend 2 3
–15
–16
_____________
bring down + 6 + 27 + 36
zeros of the divisor 3 2 9 12
_____________ 20
*multiply* • • •
remainder r(x)
_____________
coefficients of quotient q(x)
____________

4. Example #2: Let P(x) = x4 – 5x2 – 10x – 12 and d(x) = x + 2,
find
Remember to put in ___ for missing terms. 1 –5
–10
–12 –2 + 4 + 2 + 16 –2 1 –2 –1 –8 4 • • • •
Check:

5.  Example #3: Let P(x) = 2x3 + 5x2 – 7x – 12 and d(x) = x + 3, find .
Since the remainder is ___, (x + 3) is a _______ of P(x). 2 5 –7
–12
factor –6 + 3 + 12 –3 2 –1 –4 
P(x) = (______)(___________) • • •
Likewise, (–3, 0) is a _____ of the graph of y = P(x).
zero
Check:

6. CONCLUSION:
Factor Theorem
_______________: P(x) has a factor (x – k) if and only if P(x) = 0 (i.e., the remainder = ____).
Remainder Theorem
___________________: If P(x) is divided by (x – k), then the remainder = ____.

7. Using Synthetic Division to Find the Zeros of P(x)
Example #4: Given that (x – 2) and (x + 3) are factors of
P(x) = 2x4 + 7x3 – 4x2 – 27x – 18, find all the zeros of P(x).
Step 1: divide by (x – 2). 2 7 –4
–27
–18 + 4 + 22 18 + 36 + 2 2 11 18 9  • • • •
P(x) = (______)(____________________)

8. Try to factor quadratics
P(x) = (______)(____________________)
Step 2: divide __________________ by (x + 3).
2x3 + 11x2 + 18x + 9 2 11 18 9 6 5
Try to factor quadratics –6
–15 –9 3 2 –3 2 5 3  • • • x
+ 1
P(x) = (______)(______)(____________) 2x
2x2
+ 2x
+ 3
+ 3x
+ 3
P(x) = (______)(______)(______)(_______)

9. P(x) = (______)(______)(______)(_______)
Step 3: final all zeros of P(x). 2,
–3,
–1,
x = ________________
Step 4: Sketch a graph of y = P(x). y x –3 –1 2
(0, –18)

10. Work on the Synthetic Division & Finding Zeros Practice Worksheet

11. Day 10 Practice: Synthetic Division & Finding Zeros Practice
a) and 1
–12 –5 8 +2
–20
–50 +2 1
–10
–25
–42 • • •

12.  2. Given each P(x) and factors of P(x), find all the zeros of P(x).
a) , and factor (x – 2) -3 2 1 –7 6 -1 3 +2 +4 –6
P(x) = (______)(__________) +2 1 2 –3 
P(x) = (______)(______)(______) • • • x y
To find the zeros, set P(x) = 0.
0 = (______)(______)(______)
–3, 1, 2
x = _____________

13. Check your answers: 1. a) b) c) d) 2. a) x = –3, 1, 2 b) x = –4, –2, 6
c) x = –2, 1, 5

14. Finish the worksheets