The
Rational Roots Test: Introduction (page
1 of 2)

The zero of a polynomial
is an input value (usually an x-value)
that returns a value of zero for the whole polynomial when you plug it
into the polynomial. When a zero is a real (that is, not complex)
number, it is also an x-intercept
of the graph of the polynomial function. You will frequently (especially
in calculus) want to know the location of the zeroes of a given polynomial
function. You could plug numbers into the polynomial, willy-nilly, and
hope for the best. But as you learned when you studied the Quadratic
Formula, zeroes are
often very messy numbers; randomly guessing is probably not the best plan
of attack. So how does one go about trying to find zeroes?

The Rational Roots (or
Rational Zeroes) Test is a handy way of obtaining a list of useful first
guesses when you are trying to find the zeroes (roots) of a polynomial.
Given a polynomial with integer (that is, positive and negative "whole-number")
coefficients, the possible (or potential) zeroes are found by listing
the factors of the constant (last) term over the factors of the leading
coefficient, thus forming a list of fractions. This listing gives you
a list of potential
rational (fractional) roots to test -- hence the name of the Test.

Let me emphasize: The Rational
Roots Test does not
give you the zeroes. It does not say what the zeroes definitely will be.
The Test only gives you a list of relatively easy and "nice"
numbers to try in the polynomial. Most of these possible zeroes
will turn out not actually to be zeroes!

You can see the sense of
the Test's methodology by looking at a simple polynomial. Given the quadratic
12x^{2}
– 7x – 10, you
can use the Quadratic Formula to find the zeroes, but you can also factor
to get 12x^{2}
– 7x – 10 = (3x + 2)(4x – 5).
Setting the two factors equal to zero, you get two roots at x
= –^{ 2}/_{3}
and x
= ^{5}/_{4}.
Note that the denominators "3"
and "4"
are factors of the leading coefficiant "12",
and the numerators "2"
and "5"
are factors of the constant term "10".
That is, the zeroes are fractions formed of factors of the constant term
(10)
over factors of the leading coefficient (12).
Note also, however, that fractions such as
^{5}/_{6}
and –^{
10}/_{3}
may also be formed this way (and thus be provided to you by the Test),
but these other fractions are not in fact zeroes of this quadratic.

This relationship is always
true: If a polynomial has rational roots, then those roots will be fractions
of the form (plus-or-minus) (factor of the constant term) / (factor of
the leading coefficient). However, not all fractions of this form are
necessarily zeroes of the polynomial. Indeed, it may happen that none
of the fractions so formed is actually a zero of the polynomial.

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Note that I keep saying
"potential" roots, "possible" zeroes, "if there
are any such roots...". This is because the list of fractions generated
by the Rational Roots Test is just a list of potential solutions.
It need not be true that any of the fractions is actually a solution.
There might not be any fractional roots! For example, given x^{2}
– 2, the Rational Roots
Tests gives the following possible rational zeroes:

But you already know that:

...so the zeroes aren't
actually rational at all.

Always remember: The Rational
Roots Test only gives a list of good first guesses; it does NOT give you
"the" answers!

Find all possible
rational x-intercepts
of x^{4}
+ 2x^{3} – 7x^{2} – 8x + 12.

You can do a quick
graph (especially if you have a graphing calculator), and see that,
out of the above list, it would probably be good to start looking
for zeroes by plugging the values x
= –3, –2, 1, and
2
into the polynomial.