Find the product of the following binomial (2x+2y) (2x_2y)
2 answers:
Answer: (2x-2y)5
Explanation: Use the binomial expansion theorem to find each term. The binomial theorem states
(
a
+
b
)
n
=
n
∑
k
=
0
n
C
k
⋅
(
a
n
−
k
b
k
)
(
a
+
b
)
n
=
∑
k
=
0
n
n
C
k
⋅
(
a
n
-
k
b
k
)
.
5
∑
k
=
0
5
!
(
5
−
k
)
!
k
!
⋅
(
2
x
)
5
−
k
⋅
(
−
2
y
)
k
∑
k
=
0
5
5
!
(
5
-
k
)
!
k
!
⋅
(
2
x
)
5
-
k
⋅
(
-
2
y
)
k
Expand the summation.
5
!
(
5
−
0
)
!
0
!
⋅
(
2
x
)
5
−
0
⋅
(
−
2
y
)
0
+
5
!
(
5
−
1
)
!
1
!
⋅
(
2
x
)
5
−
1
⋅
(
−
2
y
)
+
…
+
5
!
(
5
−
5
)
!
5
!
⋅
(
2
x
)
5
−
5
⋅
(
−
2
y
)
5
5
!
(
5
-
0
)
!
0
!
⋅
(
2
x
)
5
-
0
⋅
(
-
2
y
)
0
+
5
!
(
5
-
1
)
!
1
!
⋅
(
2
x
)
5
-
1
⋅
(
-
2
y
)
+
…
+
5
!
(
5
-
5
)
!
5
!
⋅
(
2
x
)
5
-
5
⋅
(
-
2
y
)
5
Simplify the exponents for each term of the expansion.
1
⋅
(
2
x
)
5
⋅
(
−
2
y
)
0
+
5
⋅
(
2
x
)
4
⋅
(
−
2
y
)
+
…
+
1
⋅
(
2
x
)
0
⋅
(
−
2
y
)
5
1
⋅
(
2
x
)
5
⋅
(
-
2
y
)
0
+
5
⋅
(
2
x
)
4
⋅
(
-
2
y
)
+
…
+
1
⋅
(
2
x
)
0
⋅
(
-
2
y
)
5
Simplify the polynomial result.
32
x
5
−
160
x
4
y
+
320
x
3
y
2
−
320
x
2
y
3
+
160
x
y
4
−
32
y
5
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