Question

54x³ + 250y³

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of \(54x^3\) and \(250y^3\). The GCF of 54 and 250 is 2. So we factor out 2:
\(54x^3 + 250y^3 = 2(27x^3 + 125y^3)\)

Step2: Apply the sum of cubes formula

Notice that \(27x^3=(3x)^3\) and \(125y^3=(5y)^3\). The sum of cubes formula is \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\). Here, \(a = 3x\) and \(b = 5y\). So we can factor \(27x^3 + 125y^3\) as:
\((3x + 5y)((3x)^2 - (3x)(5y) + (5y)^2)\)
Simplify the second factor:
\((3x)^2 - (3x)(5y) + (5y)^2 = 9x^2 - 15xy + 25y^2\)

Step3: Combine the factors

Putting it all together, we have:
\(54x^3 + 250y^3 = 2(3x + 5y)(9x^2 - 15xy + 25y^2)\)

Answer:

\(2(3x + 5y)(9x^2 - 15xy + 25y^2)\)