how can the polynomial 6d^4+9d^3-12d^2 be factored?
To factor the polynomial 6d^4 + 9d^3 - 12d^2, we can observe that each term has a common factor of 3d^2:
6d^4 + 9d^3 - 12d^2 = 3d^2(2d^2 + 3d - 4)
Now, we have a quadratic expression (2d^2 + 3d - 4) remaining. To factor this quadratic, we can look for two numbers that multiply to -8 (-4 * 2) and add up to 3. The numbers are 4 and -1:
2d^2 + 3d - 4 = 3d^2 + 4d - d - 4 = d(3d + 4) - 1(3d + 4) = (3d + 4)(d - 1)
Therefore, the factored form of the polynomial 6d^4 + 9d^3 - 12d^2 is:
3d^2(3d + 4)(d - 1)