Rational Roots Test

The Rational Roots Test (also known as Rational Zeros Theorem) allows us to find all possible rational roots of a polynomial. Suppose a is root of the polynomial P\left( x \right) that means P\left( a \right) = 0. In other words, if we substitute a into the polynomial P\left( x \right) and get zero, 0, it means that the input value is a root of the function.

But how do we find the possible list of rational roots? Here’s how it works in a nutshell!


Key Ideas of Rational Roots Test

Suppose we have some polynomial P\left( x \right) with integer coefficients and a nonzero constant term:

Then every rational root of P\left( x \right) is of the form:


The best way to learn this method is to take a look at some examples!

Examples of How to Find the Rational Roots of a Polynomial using the Rational Roots Test

Example 1: Find the rational roots of the polynomial below using the Rational Roots Test.

P(x) is equal to three x cubed minus four x squared minus seventeen x plus six

Finding the rational roots (also known as rational zeroes) of a polynomial is the same as finding the rational x-intercepts.

  • Start by identifying the constant term a0 and the leading coefficient an.
  • Determine the positive and negative factors of each.

Factors of constant term, {a_0} = 6\,\,:\,\, \pm \,\left( {1,2,3,6} \right)

Factors of leading term, {a_n} = 3\,\,:\,\, \pm \,\left( {1,3} \right)

  • Write down the list of the possible rational roots by finding {p \over q} which is simply the ratio of the factors of the constant term and leading term. Make sure that you keep track of the possible combinations.

This is how I do it. I take each numerator and divide it by all denominators. Then I move on to the next numerator and again divide by all denominators. I keep repeating this process until I have gone through all the numerators. This ensures that we have covered all possible combinations.

BIG Caution: After you write down all combinations, simplify the fractions in order to get rid of duplicates.

So these are the numbers without duplicates that we will check as possible roots. We have twelve (12) possible candidates to check.

  • Remember that if a is a root of the polynomial P\left( x \right), then P\left( a \right) = 0. Now, let’s check each number.
  • Therefore, the rational roots of the polynomial
P(x)=3x^3-4x^2-17x+6

are

one-third, negative two, three

Here is the graph of the polynomial showing where it crosses or touches the x-axis. These are in fact the x-intercepts of the polynomial.


Example 2: Find the rational roots of the polynomial below using Rational Roots Test.

P(x) is equal to four x to the fifth power minus twelve to the fourth power plus 5 x cubed plus 8 x squared minus three x minus 2

The constant term is a0 = –2 and its possible factors are p = ± 1, ± 2. For the leading coefficient, we have an = 4 and its factors are q = ± 1, ± 2, ± 4.

  • To find the possible roots of the polynomial, write in the form
p over q

Write down all possible 

p/q

combinations:

  • Simplify each fraction to eliminate duplicates or identical values. Here’s our new and improved list!
  • Due to the plus or minus consideration of each number, we will have eight (8) possible candidates as the roots of this polynomial.

If you plug in each value to the given polynomial and gets zero, that means the number you substituted is a root! Try this on paper, and you should be convinced that there are only three values satisfying this condition.

Therefore, the rational roots of the polynomial

P(x)=4x^5-12x^4+5x^3+8x^2-3x-2

 are

negative one-half, one, two

Graphically, it shows that the polynomial touches or crosses the x-axis at those roots determined by rational roots test.