Polynomial function:
a. List all possible zeros
b. find all rational zeros 
c. list all factors! 
f(x) = x³ + 5x² + 2x - 8

The rule is: If a polynomial arranged in descending order
has any rational zeros, then they will be among the set 
of positive and negative fractions, each of whose numerator 
is a factor of the constant term, (i.e. the term with no
vraiable), and whose denominator is a factor of the 
coefficient of the first term.

The polynomial is so arranged.
The constant term is -8, and the coefficient of the first
term is 1, so if there are any rational zeros, they will
be among these:

±1, ±2, ±4, ±8

Try the easiest one first, 1. If 1 is a solution, then
(x - 1) will be a factor of f(x)

using synthetic division we divide by (x - 1):

1|1  5  2 -8
 |   1  6  8
  1  6  8  0

Since the last number on the bottom row of the
aynthetic division (the remainder) is 0, we
have factored f(x) as

f(x) = (x - 1)(x² + 6x + 8)

We can now factor the trinomial in the second
parentheses:

f(x) = (x - 1)(x + 2)(x + 4)

That's the list of factors, (x - 1), (x + 2), and
(x + 4)

Setting each of the factors = 0 we have

x = 1, x = -2, and x = -4 as the three zeros.

Here's the graph of f(x). Notice that it crosses
the x-axis 3 times, once at each those three zeros. 



Edwin