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Rational Root Test

Suppose we have a polynomial where the ’s represent integers which are the coefficients of the ith  power of x.  Then we say that a root of this polynomial is a value for x which makes .  For instance, the polynomial  has two roots, and , since substituting either of these into the polynomial gives a value of zero.  Sometimes finding roots of polynomials is an easy task, such as in this example, but often it is very difficult especially as the degree (highest power of the x variable) of the polynomial increases.  However we have a tool called the rational root test, which can be of assistance in some cases.

Rational Root Test:

Suppose is a polynomial with integer coefficients.  If  has a rational root of the form , then p divides and q divides .   

Remember that a rational number is a number that can be expressed as the quotient of two integers (i.e. as a fraction).  Consequently, a rational root of a polynomial P(x) as described above, is a value of x that makes P(x) = 0, where x is also a rational number.

Important!!  Note that the rational root test does not tell us when a polynomial has a rational root.  Rather, it gives us a method for listing the only possibilities for rational roots.

Here’s how it works:


Find all of the rational roots of:



First we use the rational root test to generate a list of all possible rational roots.  Then we test each one on the list to determine whether it is in fact a root.  We call the coefficient of the highest power of x the leading coefficient and the term without any x’s the constant term.  In this case, the leading coefficient is 6 and the constant coefficient is -2.  By the rational root test, if this polynomial has rational roots, , then p divides -2 and q divides 6.  So we have the following choices for p and q:

 Therefore, the candidates for rational roots are:

x =  , , , , , , ,    (all possible p/q combinations)

or more simply,

Again, it must be stressed that these are not necessarily roots of our polynomial.  In fact it may turn out that none of these is a root of our polynomial, but if our polynomial has any rational roots whatsoever, those rational roots must be one of the rational numbers in our list.  So, the next step is to begin checking each of the values.  We’ll begin at the top of the list and work our way down.

Thus the only rational roots of are and .

 It is important, however, to remember that since this polynomial is of degree 4, there are two other roots.  These roots may be irrational or they may be complex.