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Rational Zeros Calculator

Rational Zeros Calculator


Polynomial's degree

$$P(x) = a_{1}{x}^2 \pm a_{2}{x} \pm a_{3}$$

$$P(x) = a_{1}{x}^3 \pm a_{2}{x}^2 \pm a_{3}{x}\pm a_{4}$$

$$P(x) = a_{1}{x}^4 \pm a_{2}{x}^3 \pm a_{3}{x}^2 \pm a_{4}{x} \pm a_{5}$$

$$P(x) = a_{1}{x}^5 \pm a_{2}{x}^4 \pm a_{3}{x}^3 \pm a_{4}{x}^2 \pm a_{5}{x} \pm a_{6}$$

$$P(x) = a_{1}{x}^6 \pm a_{2}{x}^5 \pm a_{3}{x}^4 \pm a_{4}{x}^3 \pm a_{5}{x}^2 \pm a_{6}{x} \pm a_{7} $$

$$P(x) = a_{1}{x}^7 \pm a_{2}{x}^6 \pm a_{3}{x}^5 \pm a_{4}{x}^4 \pm a_{5}{x}^3 \pm a_{6}{x}^2 \pm a_{7}{x} \pm a_{8}$$

$$P(x) = a_{1}{x}^8 \pm a_{2}{x}^7 \pm a-{3}{x}^6 \pm a_{4}{x}^5 \pm a_{5}{x}^4 \pm a_{6}{x}^3 \pm a_{7}{x}^2 \pm a_{8}{x} \pm a_{9}$$

$$P(x) = a_{1}{x}^9 \pm a_{2}{x}^8 \pm a_{3}{x}^7 \pm a_{4}{x}^6 \pm a_{5}{x}^5 \pm a_{6}{x}^4 \pm a_{7}{x}^3 \pm a_{8}{x}^2 \pm a_{9}{x} \pm a_{10}$$












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The rational zeros calculator finds all possible rational roots of a polynomial and lets you know which of these are actual. For the polynomial you enter, the tool will apply the rational zeros theorem to validate the actual roots among all possible values.

What Is a Rational Zero?

A rational zero is a number in the form of p/q which on putting in the original polynomial yields zero.

$$ P\left(x\right) = a_{n}x^{n} + a_{n-1}x^{n-1} + … + a_{1}x + a_{0} \hspace{0.25in}where\hspace{0.25in} \left(a_{n}≠0\right) $$

Characteristics of Rational Root Theorem:

The standard rational root theorem satisfies the following conditions:

  • It gives all possible rational roots of any polynomial expression
  • Once you determine the rational zeros, the theorem will help you to verify irrational roots if any
  • The actual zeros of the polynomial will help you to graph the polynomial that lets you examine the behaviour of it

How To Find Rational Zeros?

Let us resolve an example that will help you calculate all possible and actual roots of the function given:




Factors of Constant Term:

Factors of -2 = \(+1, -1, +2, -2\) (These factors are values of “p”)

Factors of Highest Degree Term Coefficient:

Factors of 2 = \(+1, -1, +2, -2\) (These factors are values of “q”)

Possible p/q Zeros:

\(\dfrac{1}{-1}, \dfrac{-1}{-1}, \dfrac{2}{-1}, \dfrac{-2}{-1}, \dfrac{1}{2}, \dfrac{-1}{2}, \dfrac{2}{2}, \dfrac{-2}{2}, \dfrac{1}{-2}, \dfrac{-1}{-2}, \dfrac{2}{-2}, \dfrac{-2}{-2}\)

All Possible p/q Values:

\(\dfrac{1}{-1}, \dfrac{-1}{-1}, \dfrac{2}{-1}, \dfrac{-2}{-1}, \dfrac{1}{2}, \dfrac{-1}{2}, \dfrac{2}{2}, \dfrac{-2}{2}, \dfrac{1}{-2}, \dfrac{-1}{-2}, \dfrac{2}{-2}, \dfrac{-2}{-2}\)

Possible Rational Roots:

\(\dfrac{1}{2}, 1, 2, -2, -1, \dfrac{-1}{2}\)

Checking For Actual Rational Roots:

\(Root \dfrac{1}{2}:\)

\(P\left(\drac{1}{2}\right) = 2x^{6}+7x^{5}+x^{4}-3x^{3}+6x^{2}+2x-2\)

\(P\left(\drac{1}{2}\right) = 2\left(\dfrac{1}{2}\right)^{6}+7\left(\dfrac{1}{2}\right)^{5}+\left(\dfrac{1}{2}\right)^{4}-3\left(\dfrac{1}{2}\right)^{3}+6\left(\dfrac{1}{2}\right)^{2}+2\left(\dfrac{1}{2}\right)-2\)

\(P\left(\drac{1}{2}\right) = 1.25\)

\(Root 1:\)

\(P\left(1\right) = 2x^{6}+7x^{5}+x^{4}-3x^{3}+6x^{2}+2x-2\)

\(P\left(1\right) = 2\left(1\right)^{6}+7\left(1\right)^{5}+\left(1\right)^{4}-3\left(1\right)^{3}+6\left(1\right)^{2}+2\left(1\right)-2\)

\(P\left(1\right) = 13\)

\(Root 2:\)

\(P\left(2\right) = 2x^{6}+7x^{5}+x^{4}-3x^{3}+6x^{2}+2x-2\)

\(P\left(2\right) = 2\left(2\right)^{6}+7\left(2\right)^{5}+\left(2\right)^{4}-3\left(2\right)^{3}+6\left(2\right)^{2}+2\left(2\right)-2\)

\(P\left(2\right) = 370\)

\(Root -2:\)

\(P\left(-2\right) = 2x^{6}+7x^{5}+x^{4}-3x^{3}+6x^{2}+2x-2\)

\(P\left(-2\right) = 2\left(-2\right)^{6}+7\left(-2\right)^{5}+\left(-2\right)^{4}-3\left(-2\right)^{3}+6\left(-2\right)\left(-2\right)^{2}+2\left(-2\right)-2\)

\(P\left(-2\right) = -370\)

\(Root -1:\)

\(P\left(-1\right) = 2x^{6}+7x^{5}+x^{4}-3x^{3}+6x^{2}+2x-2\)

\(P\left(-1\right) = 2\left(-1\right)^{6}+7\left(-1\right)^{5}+\left(-1\right)^{4}-3\left(-1\right)^{3}+6\left(-1\right)^{2}+2\left(-1\right)-2\)

\(P\left(1\right) = -13\)

\(Root \dfrac{-1}{2}:\)

\(P\left(\drac{-1}{2}\right) = 2x^{6}+7x^{5}+x^{4}-3x^{3}+6x^{2}+2x-2\)

\(P\left(\drac{-1}{2}\right) = 2\left(\dfrac{-1}{2}\right)^{6}+7\left(\dfrac{-1}{2}\right)^{5}+\left(\dfrac{-1}{2}\right)^{4}-3\left(\dfrac{-1}{2}\right)^{3}+6\left(\dfrac{-1}{2}\right)^{2}+2\left(\dfrac{-1}{2}\right)-2\)

\(P\left(\drac{1}{2}\right) = -1.25\)

Hence proved that there exist no actual roots that fully satisfy the given polynomial. You can also put in the given statement in the rational zero theorem calculator to verify your calculations.

How Does Rational Zeros Calculator Work?

Our possible zeros calculator functions to display instant and precise calculations for rational zeros. Yes, it is possible if you follow a couple of steps:


  • Select the highest power of the polynomial
  • Enter the coefficients of all terms
  • Tap Calculate


  • The potential rational zeros calculator determines rational and actual zeros of the given polynomial


What Is The Difference Between Rational and Irrational Zeros?

A rational zero is one which has terminating decimal places in it. On the other hand, an irrational zero has non-terminating decimal places in it. With this rational zeros theorem calculator, you can not only determine all possible roots but can get a clear difference between rational and irrational roots.


From the source Wikipedia:  Rational root

From the source Khan Academy: Zeros of polynomials: matching equation to graph, Zeros with factoring

From the source Lumen Learning: Polynomial Function, The Fundamental Theorem of Algebra