Rational Root Theorem
A free College Algebra lesson from the “Polynomial Arithmetic and Theorems” unit, with a worked example and practice problems including step-by-step solutions.
The Rational Root Theorem says: every rational root of a polynomial a_n x^n + ... + a_0 (with integer coefficients) is of the form +/- p/q, where p is a factor of the constant term a_0 and q is a factor of the leading coefficient a_n. List candidates, then test with synthetic division.
What you'll learn
- List all possible rational roots of a polynomial as +/- (factor of constant) / (factor of leading coefficient)
- Test candidates with synthetic division to find actual roots
- Recognize that integer roots are a special case (when leading coefficient is 1)
Worked example
Problem. List the candidate rational roots of x^3 + 2x^2 - 5x - 6.
- Constant a_0 = -6, factors: 1, 2, 3, 6.
- Leading coefficient a_n = 1, factors: 1.
- Candidates p/q: +/- 1, +/- 2, +/- 3, +/- 6.
Answer: +/- 1, +/- 2, +/- 3, +/- 6
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. For x^2 + 3x - 4, list the positive integer candidates separated by commas (smallest first).
Show solution
- Constant -4, factors 1, 2, 4.
- Leading 1, so integer candidates 1, 2, 4.
Answer: 1,2,4
2. For x^3 - 1, list the integer candidates separated by commas (smallest positive first, then largest positive, then their negatives if needed). Just enter positive candidates.
Show solution
- Constant -1, leading 1.
- Only positive integer candidate is 1.
Answer: 1
3. For x^3 + 2x^2 - 5x - 6, list positive integer candidates separated by commas.
Show solution
- Constant -6 factors: 1, 2, 3, 6.
- Leading 1 -> integer candidates.
Answer: 1,2,3,6
4. Is x = 1 a root of x^3 + 2x^2 - 5x - 6?
Show solution
- P(1) = 1 + 2 - 5 - 6 = -8, not 0.
Answer: no
5. Is x = -1 a root of x^3 + 2x^2 - 5x - 6?
Show solution
- P(-1) = -1 + 2 + 5 - 6 = 0.
Answer: yes
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