3.2 Synthetic Division
Learning Objectives:
 Divide polynomials using synthetic division.
 Understand what synthetic division tells you about a polynomial.
 Use the Remainder Theorem to evaluated a polynomial function.
Synthetic Division
Example: Use synthetic division to find the quotient and remainder when 𝑥 3 − 4𝑥 2 − 5 is divided by 𝑥 − 3.
Step 1: _____________________
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
Step 7: Remainder: _______________________
Quotient: _________________________
Check: _________________________________
_________________________________
_________________________________
_________________________________
More Synthetic Division
You can ONLY DIVIDE by polynomials of the form _____________.
Use __________________ when terms are missing.
Example: Divide 𝑥 3 + 6𝑥 2 − 7 by 𝑥 + 3.
k=___
Interpret the result:
Recall that 3 26
1
So
𝑥 3 +6𝑥 2 −7
𝑥+3
20
=
+ 𝑥+3
Or you could write it as:
26 = 3 ⋅ 8 + 2
𝑥 2 + 6𝑥 2 − 7 = (𝑥 + 3)___________________ +_________
Another example:
3𝑥 4 −12𝑥 3 −𝑥
𝑥−2
k=___
Notice: my quotient is __________ less than the original.
Example 3: 𝑓(𝑥) =
𝑥 3 +5𝑥 2 −2𝑥−10
𝑥+5
Because the remainder is 0, we can factor:
𝑓(𝑥) =
(This also allows us to find ______________________).
The Remainder Theorem
Remainder Theorem- ______________________________________________________
_______________________________________________________________________.
What?: _________________________________________________________________
__________________________________________________________________
Example: _______________________________________________________________
2
Remainder Theorem says ___________________________________________________
Why?: ____________________________________
_____________________________________
_____________________________________
_____________________________________
Example:
Apply the Remainder Theorem to find f(3) where 𝑓(𝑥) = 2𝑥 4 − 3𝑥 3 + 𝑥 2 − 5𝑥 + 10.
Synthetic Division:
The Remainder Theorem says _____________________.
3