Question

Find all rational zeros of the polynomial.
P(x) = x^3 + 6x^2 − 32

Answer 1

A good trick whenever you have to find rational zeroes of a polynomial is to try all factors of the term without an x (In this case 32). For example, try x = 2.

Answer 2

I don't know if you're aware of it, but every polynomial can be rewritten in the following form: \[a_n x^n + a_{n-1}x^{n-1} + ... + a_1 x + a_0 = (x - x_1) \cdot (x - x_2 ) *...*(x - x_n)\] where x_1, x_2, ..., x_n are the zeroes of the polynomial. Looking at these you can see that \[a_0 = x_1 * x_2 * ... * x_n\] Hence any rational zero of p(x) must be a factor of a_0. Also, if you have found a zero, let's call it x_1, you can divide your original polynomial by (x-x_1) and the zeroes of the resulting polynomial are the remaining ones of your original polynomial