Find the product of the following binomial (2x+2y) (2x_2y)

(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