See Problem 5.
Use synthetic division and the Remainder Theorem to find P(a).
-
P
(
x
)
=
x
3
+
4
x
2
−
8
x
−
6
;
a
=
−
2
p open x close equals , x cubed , plus 4 , x squared , minus 8 x minus 6 semicolon eh equals negative 2
-
P
(
x
)
=
x
3
+
4
x
2
+
4
x
;
a
=
−
2
p open x close equals , x cubed , plus 4 , x squared , plus 4 x semicolon eh equals negative 2
-
P
(
x
)
=
x
3
−
7
x
2
+
15
x
−
9
;
a
=
3
p open x close equals , x cubed , minus , 7 x squared , plus 15 x minus 9 semicolon eh equals 3
-
P
(
x
)
=
x
3
+
7
x
2
+
4
x
;
a
=
−
2
p open x close equals , x cubed , plus 7 , x squared , plus 4 x semicolon eh equals negative 2
-
P
(
x
)
=
6
x
3
−
x
2
+
4
x
+
3
;
a
=
3
p open x close equals , 6 x cubed , minus , x squared , plus 4 x plus 3 semicolon eh equals 3
-
P
(
x
)
=
2
x
3
−
x
2
+
10
x
+
5
;
a
=
1
2
p open x close equals , 2 x cubed , minus , x squared , plus 10 x plus 5 semicolon eh equals , 1 half
-
P
(
x
)
=
2
x
3
+
4
x
2
−
10
x
−
9
;
a
=
3
p open x close equals , 2 x cubed , plus , 4 x squared , minus 10 x minus 9 semicolon eh equals 3
-
P
(
x
)
=
2
x
4
+
6
x
3
+
5
x
2
−
45
;
a
=
−
3
p open x close equals , 2 x to the fourth , plus , 6 x cubed , plus , 5 x squared , minus 45 semicolon eh equals negative 3
B Apply
-
Think About a Plan Your friend multiplies
x
+
4
x plus 4 by a quadratic polynomial and gets the result
x
3
−
3
x
2
−
24
x
+
30
.
x cubed , minus , 3 x squared , minus 24 x plus 30 . The teacher says that everything is correct except for the constant term. Find the quadratic polynomial that your friend used. What is the correct result of multiplication?
- What does the fact that all the terms except for the constant are correct tell you?
- How can polynomial division help you solve this problem?
- What is the connection between the remainder of the division and your friend's error?
-
Error Analysis A student used synthetic division to divide
x
3
−
x
2
−
2
x
x cubed , minus , x squared , minus 2 x by
x
+
1
.
x plus 1 . Describe and correct the error shown.
-
Reasoning When a polynomial is divided by
(
x
−
5
)
,
open x minus 5 close comma the quotient is
5
x
2
+
3
x
+
12
5 x squared , plus 3 x plus 12 with remainder 7. Find the polynomial.
-
Geometry The expression
1
3
(
x
3
+
5
x
2
+
8
x
+
4
)
1 third , open , x cubed , plus , 5 x squared , plus 8 x plus 4 close represents the volume of a square pyramid. The expression
x
+
1
x plus 1 represents the height of the pyramid. What expression represents the side length of the base? (Hint: The formula for the volume of a pyramid is
V
=
1
3
B
h
.
v equals , 1 third b h . )
Divide.
-
(
2
x
3
+
9
x
2
+
14
x
+
5
)
÷
(
2
x
+
1
)
open 2 , x cubed , plus , 9 x squared , plus 14 x plus 5 close divides open 2 x plus 1 close
-
(
x
4
+
3
x
2
+
x
+
4
)
÷
(
x
+
3
)
open , x to the fourth , plus , 3 x squared , plus x plus 4 close divides open x plus 3 close
-
(
x
5
+
1
)
÷
(
x
+
1
)
open , x to the fifth , plus 1 close divides open x plus 1 close
-
(
x
4
+
4
x
3
−
x
−
4
)
÷
(
x
3
−
1
)
open , x to the fourth , plus , 4 x cubed , minus x minus 4 close divides open , x cubed , minus 1 close
-
(
3
x
4
−
5
x
3
+
2
x
2
+
3
x
−
2
)
÷
(
3
x
−
2
)
open 3 , x to the fourth , minus , 5 x cubed , plus , 2 x squared , plus 3 x minus 2 close divides open 3 x minus 2 close
Determine whether each binomial is a factor of
x
3
+
x
2
−
16
x
−
16
.
x cubed , plus , x squared , minus 16 x minus 16 .
-
x
+
2
x plus 2
-
x
−
4
x minus , 4
-
x
+
1
x plus 1
-
x
−
1
x minus , 1
Use synthetic division to determine whether each binomial is a factor of
3
x
3
+
10
x
2
−
x
−
12
.
3 x cubed , plus 10 , x squared , minus x minus 12 .
-
x
+
3
x plus 3
-
x
−
1
x minus , 1
-
x
+
2
x plus 2
-
x
−
4
x minus , 4
Divide using synthetic division.
-
(
x
4
−
2
x
3
+
x
2
+
x
−
1
)
÷
(
x
−
1
)
open , x to the fourth , minus , 2 x cubed , plus , x squared , plus x minus 1 close divides open x minus 1 close
-
(
x
4
+
3
x
3
+
3
x
2
+
4
x
+
3
)
÷
(
x
+
1
)
open , x to the fourth , plus , 3 x cubed , plus , 3 x squared , plus 4 x plus 3 close divides open x plus 1 close
-
(
x
4
+
3
x
3
+
7
x
2
+
26
x
+
15
)
÷
(
x
+
3
)
open , x to the fourth , plus , 3 x cubed , plus , 7 x squared , plus 26 x plus 15 close divides open x plus 3 close
-
(
x
4
−
6
x
2
−
27
)
÷
(
x
+
2
)
open , x to the fourth , minus , 6 x squared , minus 27 close divides open x plus 2 close
-
(
x
4
−
5
x
2
+
4
x
+
12
)
÷
(
x
+
2
)
open , x to the fourth , minus , 5 x squared , plus 4 x plus 12 close divides open x plus 2 close
-
(
x
4
−
9
2
x
3
+
3
x
2
−
1
2
x
)
÷
(
x
−
1
2
)
open . x to the fourth , minus , 9 halves , x cubed , plus 3 , x squared , minus , 1 half , x . close . divides . open . x minus , 1 half . close
