5-6 The Fundamental Theorem of Algebra
Quick Review
The Fundamental Theorem of Algebra states that if
P
(
x
)
p open x close is a polynomial of degree n, where
n
≥
1
,
n greater than or equal to 1 comma then
P
(
x
)
=
0
p open x close equals 0 has exactly n roots. This includes multiple and complex roots.
Example
Use the Fundamental Theorem of Algebra to determine the number of roots for
x
4
+
2
x
2
−
3
=
0
.
x to the fourth , plus 2 , x squared , minus 3 equals 0 .
Because the polynomial is of degree 4, it has 4 roots.
Exercises
Find the number of roots for each equation.
-
x
3
−
2
x
+
5
=
0
x cubed , minus 2 x plus 5 equals 0
-
2
−
x
4
+
x
2
=
0
2 minus , x to the fourth , plus , x squared , equals 0
-
−
x
5
−
6
=
0
negative , x to the fifth , minus 6 equals 0
-
5
x
4
−
7
x
6
+
2
x
3
+
8
x
2
+
4
x
−
11
=
0
5 x to the fourth , minus , 7 x to the sixth , plus , 2 x cubed , plus , 8 x squared , plus 4 x minus 11 equals 0
Find all the zeros for each function.
-
P
(
x
)
=
x
3
+
5
x
2
−
4
x
−
2
p open x close equals , x cubed , plus 5 , x squared , minus 4 x minus 2
-
P
(
x
)
=
x
4
−
4
x
3
−
x
2
+
20
x
−
20
p open x close equals , x to the fourth , minus , 4 x cubed , minus , x squared , plus 20 x minus 20
-
P
(
x
)
=
2
x
3
−
3
x
2
+
3
x
−
2
p open x close equals , 2 x cubed , minus , 3 x squared , plus 3 x minus 2
-
P
(
x
)
=
x
4
−
4
x
3
−
16
x
2
+
21
x
+
18
p open x close equals , x to the fourth , minus , 4 x cubed , minus , 16 x squared , plus 21 x plus 18
5-7 The Binomial Theorem
Quick Review
Rows 0–5 of Pascal's Triangle are shown below.
1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
table with 6 rows and 1 column , row1 column 1 , 1 , row2 column 1 , table with 1 row and 2 columns , row1 column 1 , 1 , column 2 1 , end table , row3 column 1 , table with 1 row and 3 columns , row1 column 1 , 1 , column 2 2 , column 3 1 , end table , row4 column 1 , table with 1 row and 4 columns , row1 column 1 , 1 , column 2 3 , column 3 3 , column 4 1 , end table , row5 column 1 , table with 1 row and 5 columns , row1 column 1 , 1 , column 2 4 , column 3 6 , column 4 4 , column 5 1 , end table , row6 column 1 , table with 1 row and 6 columns , row1 column 1 , 1 , column 2 5 , column 3 10 , column 4 10 , column 5 5 , column 6 1 , end table , end table
The Binomial Theorem uses Pascal's Triangle to expand binomials. For a positive integer
n
,
(
a
+
b
)
n
=
P
0
a
n
+
P
1
a
n
−
1
b
+
P
2
a
n
−
2
b
2
+
⋯
+
P
n
−
1
a
b
n
−
1
+
P
n
b
n
,
n comma . open , eh plus b , close to the n . equals , p sub 0 , eh to the n , plus , p sub 1 . eh super n minus 1 end super . b plus , p sub 2 . eh super n minus 2 end super . b squared , plus math axis ellipsis plus . p sub n minus 1 end sub . eh . b super n minus 1 end super . plus , p sub n , b to the n , comma where
P
0
,
P
1
,
…
P
n
p sub 0 , comma , p sub 1 , comma dot dot dot , p sub n are the coefficients of the nth row of Pascal's Triangle.
Example
Use the binomial theorem to expand
(
2
x
+
3
)
3
.
open 2 x plus 3 close cubed . .
(
2
x
+
3
)
3
=
1
(
2
x
)
3
+
3
(
2
x
)
2
(
3
)
+
3
(
2
x
)
(
3
)
2
+
1
(
3
)
3
=
8
x
3
+
36
x
2
+
54
x
+
27
table with 3 rows and 1 column , row1 column 1 , open , 2 x plus 3 , close cubed , row2 column 1 , equals 1 . open , 2 x , close cubed . plus 3 . open , 2 x , close squared . open 3 close , plus 3 . open , 2 x , close . open 3 close squared . plus 1 . open 3 close cubed , row3 column 1 , equals 8 , x cubed , plus 36 , x squared , plus 54 x plus 27 , end table
Exercises
- How many numbers are in the eighth row of Pascal's Triangle?
- List the numbers in the eighth row of Pascal's Triangle.
- How many numbers are in the fifteenth row of Pascal's Triangle?
- What is the third number in the fifteenth row of Pascal's Triangle?
Use the Binomial Theorem to expand each binomial.
-
(
x
+
9
)
3
open x plus 9 close cubed
-
(
b
+
2
)
4
open b plus 2 close to the fourth
-
(
3
a
+
1
)
3
open 3 eh plus 1 close cubed
-
(
x
−
5
)
3
open x minus 5 close cubed
-
(
x
−
2
y
)
3
open x minus 2 y close cubed
-
(
3
a
+
4
b
)
5
open 3 eh plus 4 b close to the fifth
-
(
x
+
1
)
6
open x plus 1 close to the sixth
-
(
2
x
−
1
)
6
open 2 x minus 1 close to the sixth
Find the coefficient of the
x
2
x squared term in each binomial expansion.
-
(
3
x
+
4
)
3
open 3 x plus 4 close cubed
-
(
a
x
−
c
)
4
open eh x minus c close to the fourth
