Math 3
3-4 Polynomial Division & the Quotient-Remainder Theorem

Name ________________________
Let’s watch a video! http://youtu.be/hlPKliHTGZ0
In this investigation you are working towards the following learning goals:
 I can extend the relationship between standard and factored forms of polynomials
 I can perform and interpret the results of polynomial division
1.
a. How do the shape and location of the graph suggest that 𝑥 = −2 is the only zero of the function?
b. Why does the fact that 𝑐(−2) = 0 suggest that 𝑥 3 + 0𝑥 2 − 2𝑥 + 4 = (𝑥 + 2) ∙ 𝑞(𝑥) for some
polynomial 𝑞(𝑥)?
c. Why does it make sense to find the function 𝑞(𝑥) in
part (b) by calculating (𝑥 3 + 0𝑥 2 − 2𝑥 + 4) ÷ (𝑥 + 2)?
Calculating (𝑥 3 + 0𝑥 2 − 2𝑥 + 4) ÷ (𝑥 + 2) looks similar to both the polydoku and the long division
problems that we have done earlier . . .
Find (𝑥 3 + 0𝑥 2 − 2𝑥 + 4) ÷ (𝑥 + 2) using the polydoku method.
So
(𝑥 3
− 2𝑥 + 4) = (𝑥 + 2)(
x3  2 x  4

) OR
x2
Example
Find (𝑥 3 − 2𝑥 + 4) ÷ (𝑥 + 2) using long division
So (𝑥 3 − 2𝑥 + 4) = (𝑥 + 2)(
) OR
x3  2 x  4

x2
So right about now you are problem asking
“Why is writing the function 𝑥 3 − 2𝑥 + 4 in the form (𝑥 + 2)(
) so important?
Good question! We will answer that question eventually (in a later section), but for now we need to focus
on writing polynomials in equivalent forms, given one real zero. So let’s practice!
2.
a.
s  x   x3  2 x 2  4 x  7
b.
t  x   x4  2 x3  2 x2 17 x  42
SHOW ALL WORK BELOW:
a.) Zero at x = _____
b.) Zero at x = _____
x3  2 x 2  4 x  7 
x 4  2 x3  2 x 2  17 x  42 
OR
x3  2 x 2  4 x  7
OR

x 4  2 x3  2 x 2  17 x  42

3.
SHOW ALL WORK BELOW:
a.)
b.)
x3  5 x 2  4 x  20 
x3  2 x 2  4 x  7 
OR
x3  5 x 2  4 x  20

OR
x3  2 x 2  4 x  7

4. Perform the following polynomial division
a.
 3x
3
 17 x 2  10 x    x 2  5 x 
b.
 3x
5
 6 x 4  2 x3  2 x 2  3x  1   x 3  x 2  1
PROBLEM 4 CONTINUED ON NEXT PAGE 
c.
5.
 2x
5
 6 x 4  4 x 3  12 x 2  3x  7    x 2  3x 
How is the division in problem (4) different from in problem (3)?
Quotient-Remainder Theorem:
If 𝑝(𝑥) is a polynomial and 𝑑(𝑥) is a nonzero polynomial, then there exist unique polynomials 𝑞(𝑥) and
𝑟(𝑥) such that for all numbers x,
(1) 𝑝(𝑥) = 𝑞(𝑥) ∙ 𝑑(𝑥) + 𝑟(𝑥)
(2) Either degree of 𝑟(𝑥) < degree of 𝑑(𝑥)
Or 𝑟(𝑥) is the zero polynomial.
The polynomial 𝑞(𝑥) is called the quotient and 𝑟(𝑥) is called the remainder for the division of
𝑝(𝑥) by 𝑑(𝑥).