Explain the process of factoring a cubic expression.

To factor a cubic expression, we need to find its roots and use them to write the expression as a product of linear factors.

The first step in factoring a cubic expression is to check if it has any common factors that can be factored out. For example, if the expression is 2x^3 + 4x^2, we can factor out 2x^2 to get 2x^2(x+2).

If the expression does not have any common factors, we need to find its roots using various methods such as the rational root theorem, synthetic division, or the quadratic formula. Once we have found the roots, we can write the expression as a product of linear factors.

For example, consider the cubic expression x^3 - 6x^2 + 11x - 6. Using the rational root theorem, we can find that the possible rational roots are ±1, ±2, ±3, and ±6. Testing these roots, we find that x=1, x=2, and x=3 are roots of the expression. Using synthetic division, we can write the expression as (x-1)(x-2)(x-3).

In some cases, the roots may not be rational, in which case we can use the quadratic formula to find them. For example, consider the cubic expression x^3 + 3x^2 - 4x - 12. Using the quadratic formula, we can find that the roots are x=-4, x=1+√5, and x=1-√5. Therefore, we can write the expression as (x+4)(x-1-√5)(x-1+√5).

In summary, factoring a cubic expression involves finding its roots and using them to write the expression as a product of linear factors. This process can be done using various methods such as the rational root theorem, synthetic division, or the quadratic formula.