Factor completely, or state that the polynomial is prime.
375x^3 - 192y^3
375x^3 - 192y^3
= 3(125x^2 - 64y^3)
I now recognize the last factor as a difference of cubes
= 3(5x-4y)(25x^2 - 100xy + 16y^2)
To factor completely, we need to look for common factors between the terms in the polynomial.
In this case, notice that both terms have a power of 3. Thus, we can factor out the common factor of 3:
3(125x^3 - 64y^3)
Now, let's focus on the expression inside the parentheses: 125x^3 - 64y^3.
To factor this further, we can recognize that it is a difference of cubes. The difference of cubes formula is:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In this case, a is 5x and b is 4y. Applying the formula, we get:
125x^3 - 64y^3 = (5x - 4y)(25x^2 + 20xy + 16y^2)
Therefore, the completely factored form of the polynomial is:
3(5x - 4y)(25x^2 + 20xy + 16y^2)
Note: The polynomial is not prime since it can be factored further.