(c) Is
5
x
a factor of
3
2
19
10
p x
x
x
x
?
(d) Is
1
x
a factor of
3
2
7
11
3
p x
x
x
x
2
2
11
22
9
9
11 9
22
4
p x
x
x
p
(4)
For
x
a
to be a factor of
p x
, the remainder, or
p a
, must be equal to zero. This is exactly what it means for
one integers to be a factor of another integer, i.e. upon division the remainder is zero.
2
3
3
11 3
24
0
Yes,
3 is a factor
p
x
2
5
2
5
9
5
2
3
No,
5 is not a factor.
p
x
3
2
1
1
7
1
11
1
3
0
Yes,
1 is a factor
p
x
3
2
5
5
5
19 5
10
5
No,
5 is not a factor
p
x
If
4
x
is a factor of
2
52
x
kx
then
4
x
must be a zero of the same expression, i.e:
2
4
4
52
0
16
4
52
0
4
36
0
9
k
k
k
k

Name: ____________________________________
Date: __________________
C
OMMON
C
ORE
A
LGEBRA
II,
U
NIT
#10
–
P
OLYNOMIAL AND
R
ATIONAL
F
UNCTIONS
–
L
ESSON
#11
e
M
ATH
I
NSTRUCTION
,
R
ED
H
OOK
,
NY
12571,
©
2015
T
HE
R
EMAINDER
T
HEOREM
C
OMMON
C
ORE
A
LGEBRA
II
H
OMEWORK
F
LUENCY
1.
Which of the following is the remainder when the polynomial
2
5
3
x
x
is divided by the binomial
8
x
?
(1) 107
(3) 3
(2) 27
(4) 9
2. If the ratio
2
2
17
42
5
x
x
x
is placed in the form
5
r
q x
x
, where
q x
is a polynomial, then which of
the following is the correct value of
r
?
(1)
3
(3)
18
(2) 177
(4) 7
3. When the polynomial
p x
was divided by the factor
7
x
the result was
11
7
x
x
. Which of the
following is the value of
7
p
?
(1)
8
(3) 11
(2) 7
(4) It does not exist
4. Which of the following binomials is a factor of the quadratic
2
4
35
24
x
x
? Try to do this without
factoring but by using the Remainder Theorem.
(1)
6
x
(3)
8
x
(2)
4
x
(4)
2
x
5.
Which of the following linear expressions is a factor of the cubic polynomial
3
2
9
16
12
x
x
x
?
(1)
6
x
(3)
3
x
(2)
1
x
(4)
2
x
Answer Key
2
2
5
3
8
8
5 8
3
27
p x
x
x
p
(2)
2
2
2
17
42
5
2
5
17
5
42
7
p x
x
x
r
p
r
(4)
7
11
7
11
7
7
7
p x
p
q
x
x
p
x
x
x
(3)
Test the zeroes of each linear factor to see if they are
zeroes of the polynomial. Of our choices, only
8
x
is a zero, thus
8
x
is a factor.
(3)
This is the same as #4. Check the zeroes of each
linear expression to see which is a zero of the cubic.
Only
6
x
is a zero, thus
6
x
is a factor.
(1)

C
OMMON
C
ORE
A
LGEBRA
II,
U
NIT
#10
–
P
OLYNOMIAL AND
R
ATIONAL
F
UNCTIONS
–
L
ESSON
#11
e
M
ATH
I
NSTRUCTION
,
R
ED
H
OOK
,
NY
12571,
©
2015
6. Consider the cubic polynomial
3
2
46
80
p x
x
x
x
.
(a) Using polynomial long division, write the ratio of
3
p x
x
in
quotient-remainder form
, i.e. in the form
3
r
q x
x
. Evaluate
3
p
. How does this help you check your quotient-remainder form?
(b) Evaluate
5
p
. What does this tell you about the binomial