Example from start to finish
For 2x³−3x²−8x+12, p divides 12 and q divides 2, producing candidates such as ±1, ±2, ±3, ±4, ±6, ±12 and their halves. This Rational Root Theorem example can be compared with test a candidate.
Math calculator
Rational Root Theorem Calculator
List every reduced rational candidate ±p/q allowed by integer polynomial coefficients. Formula and possible rational roots remain visible in one place for independent verification.
Rational Root Theorem inputs
Calculation inputs
Mathematical foundation
Any rational root p/q in lowest terms has p dividing the constant coefficient and q dividing the leading coefficient.
Reading possible rational roots correctly starts with the mathematical structure described here.
Steps without the calculator
List positive factors of the constant and leading coefficients, form every reduced ±p/q, and remove duplicates.
Boundaries and common traps
The list contains possibilities, not guaranteed roots. A zero constant means x=0 is a root and the remaining polynomial should be treated separately. If the Rational Root Theorem assumptions do not fit, consider integer factors.
Reversing the displayed steps offers a quick independent check on this rational root theorem result.
Situations that fit the model
The theorem narrows factor searches for integer polynomials and supplies candidates for synthetic division.
Reading sensitivity and precision
The input labels—integer coefficients, highest power first—encode the model used on this page. Write those labels beside source values when transferring a problem from paper or a spreadsheet. That small step catches transposed quantities and mixed units before they become a polished-looking possible rational roots.
Larger endpoint coefficients usually expand the candidate set even when the polynomial has few actual rational roots. A one-field trial makes this relationship visible without reworking the entire example.
Choose rounding after considering how the result will be used. Comparison may need only a few significant digits, while a later multi-step calculation benefits from carrying more. In either case, retain the page's formula with the possible rational roots so the underlying definition remains visible.
The theorem proposes candidates; polynomial evaluation or synthetic division tests them. The distinction determines whether this page fits the original question.
If the number moves into a spreadsheet, give its cell a possible rational roots heading and retain the source values integer coefficients, highest power first nearby. Context matters more than extra displayed digits.
How to audit the Rational Root Theorem result
Test candidates in a deliberate order: integer candidates first, then reduced fractions. A failed candidate is still useful because it eliminates one possible linear factor.
For Rational Root Theorem, a useful audit records the formula, the supplied quantities, and one relationship that the answer must satisfy. Carry extra digits through that relationship and round only the final comparison. This separates numerical display differences from a genuine setup error.
Questions about Rational Root Theorem
Are all candidates roots?
No.
Can irrational roots occur?
Yes.
Why reduce p/q?
The theorem assumes lowest terms.
What if the constant is zero?
Zero is a root; factor out x first.