with integer coefficients (the most common situation in a math class ), and we
to find the zeros of P(x) (rational, real, or otherwise) or factor P (x). While
in reality, this
is generally impossible, here are some guide lines that will work a fair amount
of the time
in a math class.
Finding zeros is factoring. The first point is that much of the time, the
way to find the zeros of P(x) is to factor it, since if
then the zeros of P(x) are This is not
always possible, but when it works, it
works quite well.
Degree 2 is done. If P(x) has degree 2, or if you can pull linear factors
(x−c) out of
P(x) until what’s left has degree 2, then you’re done, as any polynomial of
degree 2 can be
factored using the quadratic formula .
List of guesses. In a math class, it is often (but not always) the case
that all, or almost
all, of the zeros of P(x) will be rational. Therefore, one good starting place
is to make a
list of the possible rational zeros of P(x), which, by the Rational Zeros
Theorem, is the set
of all numbers of the form , where p divides
a 0 (the constant coefficient) and q divides
an (the leading coefficient).
A step-by-step repeated procedure. With the list of possible rational zeros
we can apply the fol lowing step -by-step procedure.
1. For each possible rational zero c, calculate P (c).
2. If P(c) ≠ 0, go back to step 1 for the next c in the list of guesses, until
you run out
3. If P(c) = 0, then c is a zero of P(x), and (x−c) divides P(x). In that case,
, and start over with factoring Q(x),
possibly with a new list of guesses.
(You can eliminate any values of c that have previously been eliminated as
be careful; c may be multiple zero of P(x), so it may still be a zero of the new
If you like synthetic division, note that you can calculate
and P(c) in one
step, as P(c) is just the remainder that you get when you divide P(x) by (x −
examples in the text that follow the above procedure, see pp. 273–275.
Now, if you’re not experienced with factoring, the “start over” part of step 3
like it will take a long time. Here’s why that’s not really the case , especially
in a math class.
• If the degree of P(x) isn’t too high, because you’re basically done when you
to degree 2, it may actually only take a few steps to finish. In comparison ,
because of the ±, you might have to try lots of cases (see below).
• In a math class, it is likely that the teacher will try
to make questions reasonable, and
choose small numbers like ±1 to be the zeros.
• Just trying all of the rational zeros, without dividing, will never find any
Finding rational zeros without factoring. If you only want to find all
zeros, then one way to do that is to calculate P(c) for all values of c on the
can stop once either:
1. You find n different zeros , as P(x) can have at most n zeros; or
2. You go through all values of
The problem with this is that, for example, if P(x) has any zeros of
greater than 1, you won’t actually ever find n different zeros, which means that
end up in case 2. And that can mean trying a lot of cases: For example, if
for all rational zeros of a polynomial of the form x5 + · · · + 24, there are 16
(±1,±2,±3,±4,±6,±8,±12,±24) to check; if the polynomial has the form 7x5 + · · ·
there are 32 cases; and so on. In general, especially working by hand in a timed
factoring will probably work better.