factor the polynomial, if possible. if the polynomial cannot be factored, write "prime".
x^3+3x^2-4x-12
To factor the polynomial x^3 + 3x^2 - 4x - 12, we can begin by looking for potential rational roots using the rational root theorem. The rational root theorem states that any rational root of a polynomial equation in the form p(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x +a_0 must be of the form ± p/q, where p is a factor of the constant term a_0, and q is a factor of the leading coefficient a_n.
In this case, the constant term is -12 and the leading coefficient is 1. The factors of the constant term -12 are ±1, ±2, ±3, ±4, ±6, ±12, and the factors of the leading coefficient 1 are ±1. Thus, the potential rational roots are ±1, ±2, ±3, ±4, ±6, ±12.
By trying out these potential roots, we find that -3 is a root of the polynomial. Therefore, (x + 3) is a factor of the polynomial.
Next, we can use synthetic division to divide the polynomial by (x + 3):
-3 | 1 3 -4 -12
|_______
1 0 -4 -24
The result of the division is x^2 - 4x - 24. This is a quadratic polynomial that can be factored further.
Factoring x^2 - 4x - 24, we can find its roots by solving the equation (x - 6)(x + 4) = 0. This gives us two more potential roots: 6 and -4.
Therefore, the factored form of the polynomial x^3 + 3x^2 - 4x - 12 is (x + 3)(x - 6)(x + 4).