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
