Factor completely, or state that the polynomial is prime.
375x^3 - 192y^3
To factor the polynomial 375x^3 - 192y^3, we can use the factoring formula for the difference of cubes. The formula states that for any two numbers a and b, the difference of cubes can be factored as follows:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
In our case, the numbers are 375x^3 and 192y^3. So, we can rewrite the polynomial as:
375x^3 - 192y^3 = (5x)^3 - (6y)^3
Using the formula, we have:
(5x)^3 - (6y)^3 = (5x - 6y)((5x)^2 + (5x)(6y) + (6y)^2)
Simplifying further:
(5x - 6y)(25x^2 + 30xy + 36y^2)
Therefore, the fully factored form of the polynomial 375x^3 - 192y^3 is (5x - 6y)(25x^2 + 30xy + 36y^2).
It is important to note that factoring completely or stating that a polynomial is prime depends on the given polynomial. In this case, we were able to factor the polynomial into its simplest form.