P(x) = x^3 + 3x^2 − 4x − 12

Question

Factor the polynomial and use the factored form to find the zeros.

Answer 1

You can use either guess and check or synthetic division to find the first zero. I used guess and check and discovered that 2 is a zero, since f(2) = 0. That means (x-2) should divide evenly. You can now use either synthetic division or long division to find the quotient. I'll use synthetic division here:

1 3 -4 -12

2 | 2 10 12

----------------------

1 5 6

So we now have f(x) = (x - 2)(x² + 5x + 6). And we should recognize that the new quadratic expression is easily factorable, so we end up with:

f(x) = (x - 2)(x + 2)(x + 3)

Thus, the zeros are 2, -2, and -3.