91Ó°ÊÓ

Identify any common factors from the terms in the polynomial. In this case, we can factor out an x from all terms. Thus, the equation becomes \(x(x^{3}-13x-12)=0\)

After the first factoring, we are left with a cubic polynomial, \(x^{3}-13x-12\). To factorize this, we should look for roots of this polynomial. The roots for the cubic equation \(x^{3}-13x-12=0\) can be solved by the rational root theorem, synthetic division, or by inspection. The rational roots theorem states that if a polynomial has a rational root, \(p/q\), then p is a factor of the constant term (12 in this case) and q is a factor of the leading coefficient (1 in this case). So, from only these possibilities, -1, -12, 1, and 12 we found that the solutions are -1,1,-2,2,-3,3,-4,4,-6,6. By checking these values, we can find that -1, 1 and -12 are roots. Thus, the cubic factor can be written as \((x - 1)(x + 1)(x + 12)\). The equation becomes \(x(x - 1)(x + 1)(x + 12)=0\).

The equation is now fully factored, so we can easily find the roots by setting each factor equal to zero. We find that x = 0, x = 1, x = -1, x = -12 are solutions to the original equation.