Factor completely.
6x^4-9x^3-36x^2+54x
To factor completely, we can first factor out the greatest common factor, which in this case is 3x:
3x(2x^3 - 3x^2 - 12x + 18)
Next, we can try to factor the expression inside the parentheses. Notice that 2x^3, -3x^2, -12x, and 18 all have a common factor of 2:
3x(2x^3 - 3x^2 - 12x + 18)
= 3x(2(x^3 - (3/2)x^2 - 6x + 9))
Now, we can try to factor the expression inside the parentheses further. We can use the rational root theorem to determine possible rational roots. The possible rational roots include ±1, ±3, ±(1/2), ±(3/2), ±9.
By trying these possible values, we find that x - 1 is a factor:
2x^3 - 3x^2 - 12x + 18 = (x - 1)(2x^2 - x - 18)
Now, we can factor the quadratic expression further:
2x^2 - x - 18 = (x - 2)(2x + 9)
Therefore, the factored form of the expression is:
3x(x - 1)(x - 2)(2x + 9)