Factor completely

Question

Factor completely 48x^4y - 216x^2y + 243y

Answer 1

48x^4y-216x^2y+243y

factor out the GCF of 3y

3y(16x^4-72x^2+81)

the polynomial inside of the parentheses is a perfect square

3y(4x^2-9)(4x^2-9)

each binomial is the difference of two squares

3y(2x-3)(2x+3)(2x-3)(2x+3)

3y(2x-3)^2(2x+3)^2

Answer 2

We start with: 48x⁴y - 216x²y +243y

First, factor out the GCF which is 3y:

3y(16x⁴ - 72x² + 81)

Which can be rewritten as:

3y((4²)(x²)² - 2(4)(9)x² + 9²)

The part in the parentheses now follows the formula for a perfect square trinomial:

a²w² ± 2abw + b² = (aw ± b)² with

a = 4

b = 9

w = x²

So, your expression factors to:

48x⁴y - 216x²y +243y = 3y(16x⁴ - 72x² + 81) = 3y((4²)(x²)² - 2(4)(9)x² + 9²) = 3y(4x² - 9)²

3y(4x² - 9)² is your final answer.