Question

State how many imaginary and real zeros the function has. f(x) = x3 + 5x2 - 28x - 32

Answer 1

All equation has always even numbers of imaginary roots because they always come in pairs. Therefore there must be either 2 imaginary roots, or zero.

To determine the types of roots, we could try synthetic division by some of the factors of the number -32. The simplest factors are +1 or -1. Using this, we can quickly show that (x + 1) is a factor, and leaving (x² + 4x - 32) as the remainder. This remainder equation then factorises to (x - 4) (x + 8).

So to conclude there are three real zeros which are:

x = -1

x = 4

x = -8