How do you factor an x^3 as in x^3+5x^2+6? Is there a general rule you can follow?

To factor a cubic polynomial, like x^3+5x^2+6, you generally need to use a combination of techniques such as factoring by grouping, using the factor theorem, or using synthetic division.

In this particular case, let's apply the factoring by grouping method:

Step 1: Group the terms.
Take the first two terms, x^3 and 5x^2, and the last term, 6, and pair them up:
(x^3 + 5x^2) + 6

Step 2: Look for common factors.
In the first pair, x^3 and 5x^2, the highest common factor is x^2. So, we can factor out x^2:
x^2(x + 5) + 6

Step 3: Factor the remaining expression.
Now, we have x^2(x + 5) + 6, which cannot be further factored using simple integers. Therefore, we stop here.

The factored form of x^3+5x^2+6 is x^2(x + 5) + 6.

Please note that factoring is not guaranteed for every cubic polynomial. In some cases, more advanced techniques such as the rational root theorem, synthetic division, or even numerical methods might be required.