Rational Zeros Theorem Calculator

Find all possible rational zeros of polynomials step by step

The calculator will find all possible rational roots of the polynomial using the rational zeros theorem. After this, it will decide which possible roots are actually the roots. This is a more general case of the integer (integral) root theorem (when the leading coefficient is $$$1$$$ or $$$-1$$$). Steps are available.

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Our Rational Zeros Theorem Calculator is a great tool to simplify finding the potential rational zeros for a polynomial function. With powerful algorithms under its hood, this calculator efficiently computes all potential and actual rational zeros.

How to Use the Rational Zeros Theorem Calculator?

  • Input

    Input the given polynomial in the specified field.

  • Calculation

    After ensuring the polynomial is entered correctly, click on the "Calculate" button.

  • Result

    Based on the Rational Zeros Theorem, the calculator will quickly display potential and actual rational zeros.

What Is a Rational Zero?

In the context of polynomial functions, a rational zero refers to a root (or x-intercept) of the polynomial that is a rational number. A rational number is any number that can be expressed as the quotient or fraction $$$\frac{p}{q}$$$ of two integers, where $$$p$$$ is the numerator, $$$q$$$ is the non-zero denominator.

To clarify:

  • Polynomial Function: A mathematical expression involving a sum of powers of one or more variables multiplied by some coefficients. For instance, $$$f(x)=x^3-4x^2+2x-8$$$ is a polynomial.
  • Rational Number: Numbers that can be written as a fraction. For example, $$$0.75$$$, which can be written as $$$\frac{3}{4}$$$, and $$$5$$$, which can be written as $$$\frac{5}{1}$$$, are rational numbers.
  • Rational Zero: The values at which the polynomial is zero are called the zeros or roots of the polynomial. If any of these zeros can be expressed as a fraction of integers, they are called rational zeros.

The Rational Zeros Theorem is a helpful tool in polynomial algebra, assisting in determining potential rational zeros of a polynomial.

How to Use the Rational Zeros Theorem?

The Rational Zeros Theorem provides a method to determine all possible rational zeros (or roots) of a polynomial function. Here's how to use the theorem:

  • Identify Coefficients: Note a polynomial's leading coefficient and the constant term. For example, in $$$f(x)=3x^3-4x^2+2x-6$$$ they are $$$3$$$ and $$$-6$$$.
  • List Factors: Enumerate the factors of both the constant term and the leading coefficient: $$$-1,1,-2,2,-3,3,-6,6$$$.
  • Calculate Potential Zeros: Create fractions using each factor of the constant term over each factor of the leading coefficient. For our example, the potential zeros are $$$\pm1,\pm2,\pm3,\pm6,\pm\frac{1}{3}\pm\frac{2}{3}$$$.
  • Verify Zeros: Use synthetic division or substitution to check each potential zero. If the polynomial equals zero for some value, that's a rational zero.

Remember, the Rational Zeros Theorem provides all possible rational zeros. The number of actual rational zeros will be fewer (or even no rational zeros at all), and there may also be irrational or complex zeros. Also, the Rational Zeros Theorem is applicable only when all coefficients of the polynomial are integers.

Why Choose Our Rational Zeros Theorem Calculator?

  • Precision and Accuracy

    Our tool ensures that every result is both precise and correct.

  • User-Friendly Design

    Even if you're new to the concept, the intuitive interface makes it simple to input your data and get results without any hassle.

  • Speed

    Eliminate manual calculations and errors. Our calculator delivers instant results, saving you time.

  • Educational Value

    Beyond just providing answers, our calculator aids in understanding the theorem's application, enhancing the learning process.

  • Compatibility

    Whether you're on a desktop, tablet, or mobile, our calculator is optimized for all devices, ensuring a smooth experience every time.


How is the Rational Zeros Theorem helpful?

The theorem provides a systematic way to identify all potential rational zeros of a polynomial, offering a starting point for polynomial factoring and solution finding.

What is the purpose of the Rational Zeros Theorem Calculator?

The calculator helps to determine all potential rational zeros of a polynomial, streamlining the process and ensuring accuracy.

Can I use the calculator for polynomials of any degree?

The calculator is designed to handle polynomials of different degrees. Just ensure you input all the coefficients correctly.

Does the calculator handle both positive and negative coefficients?

Absolutely. The calculator can process both positive and negative coefficients without any issues.