How can the polynomial 6d^4 + 9d³ – 12d² be factored?

Answer 1

First, we can factor out the greatest common factor, which is 3d². This gives us:

3d²(2d² + 3d - 4)

Next, we need to factor the quadratic trinomial 2d² + 3d - 4. We can do this by finding two numbers that multiply to -8 (the product of 2 and -4) and add up to 3 (the coefficient of the middle term). Those two numbers are 4 and -1. So we can rewrite the quadratic trinomial as:

3d²(2d² + 4d - d - 4)

Now, we can factor by grouping:

3d²(2d(d + 2) - 1(d + 2))

Finally, we can factor out the common factor (d + 2) from both terms:

3d²(d + 2)(2d - 1)

Therefore, the polynomial 6d^4 + 9d³ - 12d² can be factored as 3d²(d + 2)(2d - 1).