Rational Root Theorem
Possible rational roots are factors of the constant over factors of the leading coefficient.
About this topic
The Rational Root Theorem narrows the search for roots of a polynomial with integer coefficients. Any rational root p/q must have p dividing the constant term and q dividing the leading coefficient. For 2x³ − x² − 7x + 6, candidates include ±1, ±2, ±3, ±6, ±1/2 and ±3/2.
Without it you would be guessing among infinitely many numbers. With it you have a finite list to test, using substitution or synthetic division. Testing x = 1 gives 2 − 1 − 7 + 6 = 0, so 1 is a root, and dividing leaves 2x² + x − 6 = (2x − 3)(x + 2), so the roots are 1, 3/2 and −2.
Finding roots helps solve cubic and higher equations that appear in volume problems, engineering designs and optimization, where no simple formula exists.
The theorem only lists possible roots; most will fail, and the actual roots may be irrational or complex, which this method cannot find. Remember to include both positive and negative candidates. After you find one root, divide it out and solve the smaller polynomial, which is usually a quadratic you can factor or handle with the formula.
Key ideas
- List ±p/q candidates
- Test with synthetic division
- Reduce the polynomial after each root
Quick check
Use cases: where Rational Root Theorem helps
- Homework: review the key idea behind Rational Root Theorem before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Rational Root Theorem without notes.
- Visual learning: play with the interactive lab to see how Rational Root Theorem changes when you adjust its inputs.
- Teaching a friend: explain Rational Root Theorem in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Rational Root Theorem?
Possible rational roots are factors of the constant over factors of the leading coefficient.
What will I learn in Rational Root Theorem?
List ±p/q candidates. Test with synthetic division. Reduce the polynomial after each root.
Is Rational Root Theorem part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Rational Root Theorem lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Rational Root Theorem?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.