How to Factor Polynomials and Find Their Zeros

Suppose we have a polynomial



with integer coefficients (the most common situation in a math class ), and we either want
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 most efficient
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 in hand,
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
of guesses.
3. If P(c) = 0, then c is a zero of P(x), and (x−c) divides P(x). In that case, calculate
, 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 zeros, but
be careful; c may be multiple zero of P(x), so it may still be a zero of the new quotient
Q(x).)

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 − c). For
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 may look
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 reduce
to degree 2, it may actually only take a few steps to finish. In comparison , especially
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 non-rational
zeros.

Finding rational zeros without factoring. If you only want to find all rational
zeros, then one way to do that is to calculate P(c) for all values of c on the list. You
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 multiplicity
greater than 1, you won’t actually ever find n different zeros, which means that you’ll
end up in case 2. And that can mean trying a lot of cases: For example, if you’re looking
for all rational zeros of a polynomial of the form x⁵ + · · · + 24, there are 16 cases
(±1,±2,±3,±4,±6,±8,±12,±24) to check; if the polynomial has the form 7x⁵ + · · · + 24,
there are 32 cases; and so on. In general, especially working by hand in a timed situation,
factoring will probably work better.