Rational Zeros Calculator

Rational Zeros Calculator

Enter the polynomial in the designated field, and the calculator will calculate all possible rational roots. It will also determine which of these zeros satisfy the entered polynomial.

Polynomial's degree

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

a₁:

a₂:

a₃:

Rational Zeros Calculator

Use our rational zeros calculator to find all possible rational roots of a polynomial and determine which of these are actual zeros. This tool applies the Rational Root Theorem to verify real roots among all possible candidates quickly and accurately.

What Is a Rational Zero?

A rational zero of a polynomial is a number of the form p/q that, when substituted into the polynomial, yields zero.

For a polynomial:

\( P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \), where \( a_n \ne 0 \)

Rational Root Theorem Characteristics

The Rational Root Theorem helps in the following ways:

Lists all possible rational roots of any polynomial expression.
Helps verify irrational roots once rational zeros are determined.
The actual zeros can be used to graph the polynomial and analyze its behavior.

How to Find Rational Zeros

Example Polynomial:

\( 2x^{6} + 7x^{5} + x^{4} - 3x^{3} + 6x^{2} + 2x - 2 \)

Step 1: Factors of the Constant Term

Factors of -2: \( \pm 1, \pm 2 \) (possible values for p)

Step 2: Factors of Leading Coefficient

Factors of 2: \( \pm 1, \pm 2 \) (possible values for q)

Step 3: Possible Rational Zeros (p/q)

\( \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{2}{2} \)

Step 4: Check Each Candidate

Root 1/2: \( P(1/2) = 1.25 \) → not a zero
Root 1: \( P(1) = 13 \) → not a zero
Root 2: \( P(2) = 370 \) → not a zero
Root -2: \( P(-2) = -370 \) → not a zero
Root -1: \( P(-1) = 1 \) → not a zero
Root -1/2: \( P(-1/2) = -1.25 \) → not a zero

Hence, this polynomial has no actual rational zeros. Verification can also be done using the rational zeros theorem calculator.

How the Rational Zeros Calculator Works

Input:

Select the highest power of the polynomial
Enter the coefficients of all terms
Click "Calculate"

Output:

The calculator lists all possible rational zeros
Identifies which candidates are actual zeros of the polynomial

FAQs

What is the difference between rational and irrational zeros?

A rational zero has a terminating or repeating decimal, while an irrational zero has a non-terminating, non-repeating decimal. Our calculator distinguishes between all possible rational and irrational roots easily.

Rational Zeros Calculator

Rational Zeros Calculator

What Is a Rational Zero?

Rational Root Theorem Characteristics

How to Find Rational Zeros

Example Polynomial:

Step 1: Factors of the Constant Term

Step 2: Factors of Leading Coefficient

Step 3: Possible Rational Zeros (p/q)

Step 4: Check Each Candidate

How the Rational Zeros Calculator Works

Input:

Output:

FAQs

What is the difference between rational and irrational zeros?

References