Question

Factorize : a^6 - 18a^3 + 125. Its very difficult for me.

Answer 1

18a^3 is taken as 45a^3 - 27a^3

Hence, the given one is = a^6 - 27a^3 + 125 + 45a^3

= (a²)^3 + (-3a)^3 + (5)^3 - 3(a²)(-3a)(5)

Now, we can apply, a^3 + b^3 + c^3 - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Here a = a²; b = -3a and c = 5

Thus, a^6 + 18a^3 + 125 = (a²)^3 + (-3a)^3 + (5)^3 - 3(a²)(-3a)(5)

= (a² - 3a + 5){(a^4) + 9a² + 25 + 3(a^3) + 15a - 5a²}

= (a² - 3a + 5){(a^4) + 3(a^3) + 4a² + 15a + 25}