Verified answer

1st factor (2y):

= 32x²y - 2y³

= 2y(16x² - y²)

2nd & 3rd factors:

16x² - y² = 0

16x² = y²

x² = 1/16y²

x = ± 1/4y

= x - 1/4y, = 4x - y

= x + 1/4y, = 4x + y

Answer: 2y(4x - y)(4x + y)

Checking:

= 2y(4x - y)(4x + y)

= 2y(4x[4x] + 4x[y] - y[4x] - y[y])

= 2y(16x² + 4xy - 4xy - y²)

= 2y(16x² - y²)

= 2y(16x²) + 2y(- y)²

= 32x²y - 2y³

32x²y - 2y³ =

Look at each term's factors.

32x²y = 1 * 2 * 2 * 2 * 2 * 2 * x * x * y

-2y³ = -1 * 1 * 2 * y * y * y

What they have in common is the GCF.

GCF = 1 * 2 * y = 2y

Factor that out.

32x²y - 2y³ =

2y(16x² - y²) =

Notice that you have the difference of two perfect squares inside the parentheses.

16x² = 4x * 4x = (4x)²

y² = (y)²

16x² - y² = (4x)² - (y)²

This means:

2y(16x² - y²) =

2y[(4x)² - (y)²] =

Remember how to factor the difference of two squares.

a² - b² = (a - b)(a + b)

Apply this to what you have.

2y[(4x)² - (y)²] =

2y(4x - y)(4x + y)

ANSWER: 2y(4x - y)(4x + y)