Question

3x^3+6x^2y-y^3 factorise ​

Answer 1

Answer:

The way it is given to you, it is almost obvious that it is (3x-y)x² + (3x-y)2xy + (3x-y)y² = (3x-y)(x²+2xy+y²) = (3x-y)(x+y)².

However, if you don’t notice that and just sum like terms together, you get 3x³ + 5x²y + xy² - y³ = -x³(t³ - t² - 5t - 3), where t=y/x. Now the roots of this cubic polynomial in t can easily be found: trying the divisors of the constant term, we see that t=-1 and t=3 are roots. Therefore t³ - t² - 5t - 3 is divisible by t-3, and actually carrying out the ddivision we see that t³ - t² - 5t - 3 = (t-3)(t²+2t+1) = (t-3)(t+1)², whence 3x³ + 5x²y + xy² - y³ = (y-3x)(y+x)² once again.