300y^9z^6-4x^6+43y^9z^6

Question

Factor the polynomial

Answer 1

300y⁹z⁶ - 4x⁶ + 43y⁹z⁶

300y⁹z⁶ + 43y⁹z⁶ - 4x⁶

343y⁹z⁶ - 4x⁶

7³y⁹z⁶ - 4x⁶

(7y³z²)³ - 4x⁶

This would be MY final answer since, although most everything is a perfect cube, the "4" is not.

If the "4" were a perfect cube, you could factor this as the difference of 2 cubes. Since it's not, you can't factor it without breaking the 4 into cube roots. I'm unsure if that what your teacher is looking for but I know I would not be looking for that if I were the teacher. If by chance they were, you could do it as follows:

Difference of two cubes: a³ – b³ = (a – b)(a² + ab + b²)

a = 7y³z² and b = cuberoot(4)x²

So the factors are (7y³z² - cuberoot(4)x²)(49y⁶z⁴ + 7cuberoot(4)x²y³z² + cuberoot(16)x⁴)