Lesson 2.14
The Rational Root Theorem
Stuck trying to find a root? The Rational Root Theorem gives you a finite list of candidates to test — drastically narrowing the search from infinity to a manageable checklist.
Introduction
You know from the Factor Theorem that if , then is a factor. But how do you find that in the first place? The Rational Root Theorem tells you every possible rational number that could be a zero.
Past Knowledge
Factor Theorem and synthetic division (Lessons 2.8, 2.13).
Today's Goal
Generate a list of possible rational roots, test them, and use the survivors to factor completely.
Future Success
Next chapter: combine this with quadratic formula to find all zeros — real and imaginary.
Key Concepts
The Theorem
If has integer coefficients, then every rational zero has the form:
where is a factor of the constant term and is a factor of the leading coefficient .
The Process
💡 Key Insight
The theorem only lists possible candidates. Not all of them will be actual roots — you still have to test each one.
Worked Examples
Example 1: Leading Coefficient = 1
BasicFind all rational zeros of .
List Possible Rational Roots
Factors of constant :
Factors of leading coeff :
Candidates :
Test
| 1 | 1 | −6 | 11 | −6 |
| 1 | −5 | 6 | ||
| 1 | −5 | 6 | 0 |
✓ Remainder is 0 — is a root!
Factor the Quotient
Zeros:
Example 2: Leading Coefficient ≠ 1
IntermediateFind all rational zeros of .
List Possible Rational Roots
(factors of ):
(factors of ):
Candidates:
Try First
| 1 | 2 | −3 | −8 | −3 |
| 2 | −1 | −9 | ||
| 2 | −1 | −9 | −12 |
✗ Remainder is −12 — is NOT a root. Move on to the next candidate.
Try
| 3 | 2 | −3 | −8 | −3 |
| 6 | 9 | 3 | ||
| 2 | 3 | 1 | 0 |
✓ Remainder is 0 — is a root! Quotient:
Factor the Quotient
Zeros:
Example 3: No Rational Roots
AdvancedFind all rational zeros of .
List Possible Rational Roots
Candidates: (only possibilities when both and )
Test Both Candidates
≠ 0 ✗
≠ 0 ✗
Conclusion
This polynomial has no rational roots.
Its one real root is irrational. You'd need numerical methods or the cubic formula to find it.
Common Pitfalls
Forgetting the ± Signs
Every factor of and includes both positive and negative versions. Don't list only the positive candidates.
Thinking Every Candidate Is a Root
The theorem only gives possible rational roots. Most candidates will fail. You must test each one with substitution or synthetic division.
Real-Life Applications
Computer graphics engines use polynomial root-finding to determine where a ray of light intersects a curved surface. The Rational Root Theorem provides a first pass to narrow down intersection points — if the surface is defined by an equation with integer coefficients, the theorem identifies all "nice" intersection points before numeric methods refine the rest.
Practice Quiz
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