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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

Why it matters: Engineering, physics, and computer-algebra systems all need to factor polynomials over the rationals before working over the reals or complex numbers — the rational root theorem is the standard first step.

Worked example

Problem. List the candidate rational roots of x^3 + 2x^2 - 5x - 6.

  1. Constant a_0 = -6, factors: 1, 2, 3, 6.
  2. Leading coefficient a_n = 1, factors: 1.
  3. Candidates p/q: +/- 1, +/- 2, +/- 3, +/- 6.

Answer: +/- 1, +/- 2, +/- 3, +/- 6

Practice problems

1. For x^2 + 3x - 4, list the positive integer candidates separated by commas (smallest first).

Show solution
  1. Warm-up: First identify exactly what the question is asking: For x^2 + 3x - 4, list the positive integer candidates separated by commas (smallest first).
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Constant -4, factors 1, 2, 4.
  4. Leading 1, so integer candidates 1, 2, 4.
  5. Check the result by substituting or estimating: the response should match 1,2,4 and make sense in the original problem.

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
  1. Warm-up: First identify exactly what the question is asking: 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.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Constant -1, leading 1.
  4. Only positive integer candidate is 1.
  5. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

3. For x^3 + 2x^2 - 5x - 6, list positive integer candidates separated by commas.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For x^3 + 2x^2 - 5x - 6, list positive integer candidates separated by commas.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Constant -6 factors: 1, 2, 3, 6.
  4. Leading 1 -> integer candidates.
  5. Check the result by substituting or estimating: the response should match 1,2,3,6 and make sense in the original problem.

Answer: 1,2,3,6

4. Is x = 1 a root of x^3 + 2x^2 - 5x - 6?

Show solution
  1. Core Practice: First identify exactly what the question is asking: Is x = 1 a root of x^3 + 2x^2 - 5x - 6?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. P(1) = 1 + 2 - 5 - 6 = -8, not 0.
  4. Check the result by substituting or estimating: the response should match no and make sense in the original problem.

Answer: no

5. Is x = -1 a root of x^3 + 2x^2 - 5x - 6?

Show solution
  1. Core Practice: First identify exactly what the question is asking: Is x = -1 a root of x^3 + 2x^2 - 5x - 6?
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. P(-1) = -1 + 2 + 5 - 6 = 0.
  4. Check the result by substituting or estimating: the response should match yes and make sense in the original problem.

Answer: yes

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