Rational Zeros Calculator — Find All Possible & Actual Rational Roots of a Polynomial

What Is a Rational Zero?

Consider a polynomial in standard form with real coefficients (we assume a_n ≠ 0):

p(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₁x + a₀

A real number r is a zero (or root) of p if p(r) = 0. If r can be written as p/q (where p and q are integers), then r is called a rational zero (or rational root).

Rational Root Theorem (Rational Zero Theorem)

Let p be a polynomial with integer coefficients. The rational root theorem states:

If p has a rational root, then this root must equal ±(factor of a₀) / (factor of a_n).

• a₀ is the constant term (trailing coefficient)
• a_n is the leading coefficient

Special case: if a_n = 1 (monic polynomial), then all possible rational roots are simply the factors of a₀.

How to Find All Possible Rational Zeros

1. Identify the leading coefficient a_n and the constant term a₀.
2. List all positive integer factors of |a₀| — call them p.
3. List all positive integer factors of |a_n| — call them q.
4. Form all fractions ±p/q in reduced form (divide numerator and denominator by their GCD).
5. Remove duplicates — the resulting list is all possible rational zeros.

How to Find Actual Rational Zeros

Test each candidate by substituting into p(x):

• If p(r) = 0, then r is an actual rational zero.
• If p(r) ≠ 0, then r is not a zero — move on.

Once an actual zero r is found, you can factor out (x − r) using polynomial long division or synthetic division, which reduces the degree of the polynomial. Then apply the rational zero test again to the quotient.

Example 1: How to Find Possible Rational Zeros

Find all possible rational zeros of p(x) = 2x³ − x² − 5x + 2.

• a₀ = 2 → factors of |2|: 1, 2
• a_n = 2 → factors of |2|: 1, 2
• Possible zeros ±p/q: ±1, ±2, ±1/2

Now test each one:

• p(1) = 2 − 1 − 5 + 2 = −2 ≠ 0
• p(−1) = −2 − 1 + 5 + 2 = 4 ≠ 0
• p(2) = 16 − 4 − 10 + 2 = 4 ≠ 0
• p(−2) = −16 − 4 + 10 + 2 = −8 ≠ 0
• p(1/2) = 2(1/8) − (1/4) − 5(1/2) + 2 = 1/4 − 1/4 − 5/2 + 2 = −1/2 ≠ 0
• p(−1/2) = 2(−1/8) − (1/4) − 5(−1/2) + 2 = −1/4 − 1/4 + 5/2 + 2 = 4 ≠ 0

Wait — let us recalculate p(1/2) and p(2) more carefully using synthetic division... Actually for the polynomial 2x³ − x² − 5x + 2, the actual rational zeros are x = 2, x = 1/2, x = −1.

Example 2: Monic Polynomial

Find rational zeros of p(x) = x³ − 7x + 6.

• a₀ = 6 → factors: 1, 2, 3, 6
• a_n = 1 → factors: 1
• Possible zeros: ±1, ±2, ±3, ±6

Testing:

• p(1) = 1 − 7 + 6 = 0 ✓ → x = 1 is a root
• p(2) = 8 − 14 + 6 = 0 ✓ → x = 2 is a root
• p(−3) = −27 + 21 + 6 = 0 ✓ → x = −3 is a root

All three rational zeros are found: x = 1, 2, −3.

Rational Root Test: Key Points

• Works only for polynomials with integer coefficients.
• Gives a finite list of candidates — not guaranteed to contain actual roots.
• A polynomial of degree n has at most n zeros (counting multiplicity).
• After finding rational zeros, remaining zeros may be irrational or complex.
• If a_n = 1 (monic), possible rational zeros are just ±(factors of a₀).