When the polynomial is degree 3 or higher, use your tools to make
good guesses. When you find a zero, divide it out, work on a
polynomial of lower degree. Good Guessing and Good Luck are key.
Factor a
Factor a
polynomial
polynomial of
of degree 𝒏
degree 𝒏 − 𝟏
1. Find a zero at 𝒙 = 𝒄.
2. Divide the polynomial by (𝒙 − 𝒄).
3. You now have a new polynomial of lower
degree to work with.
Note: If the zero is a complex zero or an irrational zero,
its conjugate is a zero, too. So divide them both out in
one step: (𝑥 − 𝑐)(𝑥 + 𝑐) → 𝑥 2 − 𝑐 2 .
You have several tools to help with Good Guessing:
 The Rational Zero Theorem
 Descartes’s Rule of Signs
 The Bounded Zero Theorem
 Polynomial Long Division
 Synthetic Division
Repeat as you
ratchet down one
degree at a time,
obtaining simpler
polynomials of
lower degree at
each step.
Factor a
polynomial
of degree 𝟑
Then when you get it down
to degree 2, you’re on solid
ground! The end is near!
Factor a polynomial of degree 𝟐
1. Find a zero at
𝑥 = 𝑐.
2. Divide the
polynomial by
(𝑥 − 𝑐).
3. Now you have
degree 2.
There are only two more zeros to
find and you should have 100%
confidence about being able to find
these last two zeros!
1. Factoring as usual like you
did in basic Algebra class.
2. If you can’t factor it, use
the Quadratic Formula.
and some informal tools such as
 Good intuition and lucky guessing
 TI-84 graphing, CALC, 2:zero (when
permitted)
 TI-84 computational assistance
(But the TI-89 and certain web sites are not in
keeping with the spirit of the exercise and
should be avoided.)
The goal is to get down to Degree 2,
where we have familiar methods
that lead to a definite solution
without any trial-and-error or
guesswork.
Some more quick tricks that often work


When you find that 𝑐 is a zero of 𝑃(𝑥), it might have multiplicity of 2 or more. That is, after you divide out 𝑥 − 𝑐 and get a new 𝑃(𝑥),
plug in to calculate 𝑃(𝑐) and if that’s zero, then divide out 𝑥 − 𝑐 again.
Good initial guesses for zeros are 𝑐 = 1, 𝑐 = −1, 𝑐 = 2, 𝑐 = −2, etc. Plug in some “small” values like that (perhaps first checking to see
if the Rational Zero Theorem says they’re possibly zeros). A lot of problems use polynomials which have these small integers among
their zeros.
Document1
2/10/2016 6:07 AM - D.R.S.