Analyze rational candidates using factor pairs and exact substitution. See reduced fractions and confirmed roots. Download tables and plots for classwork, checking, or revision.
| Polynomial | p Factors | q Factors | Candidates | Confirmed Rational Roots |
|---|---|---|---|---|
| x3 - 6x2 + 11x - 6 | 1, 2, 3, 6 | 1 | ±1, ±2, ±3, ±6 | 1, 2, 3 |
| 2x3 - x2 - 8x + 4 | 1, 2, 4 | 1, 2 | ±1, ±2, ±4, ±1/2 | -2, 1/2, 2 |
The calculator uses the Rational Root Theorem. For a polynomial written as anxn + an-1xn-1 + ... + a0, any rational root in lowest form is written as p/q.
Here, p must divide the constant term a0. Also, q must divide the leading coefficient an. The calculator generates every reduced candidate ±p/q, then substitutes each value into the polynomial.
If the exact evaluation becomes zero, that candidate is a confirmed rational root. When the constant term is zero, the calculator also detects zero roots by removing trailing zero coefficients first.
P over q rational root testing helps you narrow down possible rational zeros before using factorization, synthetic division, or graph analysis. It is especially useful for integer-coefficient polynomials where blind guessing wastes time.
The method begins with the constant term and the leading coefficient. Every factor of the constant can serve as p. Every factor of the leading term can serve as q. Reduced fractions formed from these values create the complete rational candidate set.
This calculator automates that entire workflow. It accepts an ordered coefficient list, builds candidate fractions, reduces duplicates, evaluates each one exactly, and separates confirmed rational roots from rejected options. That saves repeated hand substitution and makes checking easier.
The result section also highlights zero-root multiplicity when the polynomial ends with one or more zero coefficients. That matters because x itself becomes a factor, and the reduced polynomial can then be tested again using the same theorem.
The graph adds visual support. Confirmed rational roots sit on the x-axis, while the polynomial curve shows turning behavior, crossings, and nearby growth. This makes the calculator helpful for homework checking, algebra revision, and exam preparation.
It represents any possible rational root in lowest form. The numerator p comes from factors of the constant term, and denominator q comes from factors of the leading coefficient.
Yes. The Rational Root Theorem is stated for polynomials with integer coefficients. That is why this calculator validates integer input for the coefficient list.
Reduced fractions remove duplicates. For example, 2/4 and 1/2 represent the same candidate. Keeping lowest terms gives a clean, complete candidate list.
No. It only lists possible rational roots. Irrational and complex roots can still exist, even when no p over q candidate gives zero.
If the constant term is zero, x is a factor. That means zero is a root. Repeated trailing zeros can also show repeated zero-root multiplicity.
It shows the substituted result for each candidate fraction. When the exact result is zero, that candidate is confirmed as a rational root.
Yes. The calculator works for any polynomial degree as long as you provide valid integer coefficients and a nonzero leading coefficient.
The graph helps you see where the polynomial crosses or touches the x-axis. It also shows overall shape and supports quick visual checking.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.