Rational Zeros Calculator

The underlying mathematics relies entirely on the Rational Root Theorem. Given a polynomial equation of the form P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, any rational root expressed in lowest terms as p/q must satisfy specific divisibility conditions. Specifically, the numerator p must be an integer factor of the constant term a₀, and the denominator q must be an integer factor of the leading coefficient a_n. The calculator automatically generates all possible combinations of ±(factors of a₀) / (factors of a_n) and evaluates synthetic division to confirm actual roots.

Let us find the rational zeros of the polynomial P(x) = 2x³ - 3x² - 8x - 3. First, identify the constant term a₀ = -3 and the leading coefficient a_n = 2. The integer factors of the constant term (-3) are ±1 and ±3. The integer factors of the leading coefficient (2) are ±1 and ±2. Next, construct all potential rational roots by dividing every factor of a₀ by every factor of a_n, yielding the set: ±1, ±3, ±1/2, and ±3/2. Testing these values using synthetic division or direct substitution reveals that x = 3 is an actual zero, because P(3) = 2(27) - 3(9) - 8(3) - 3 = 54 - 27 - 24 - 3 = 0. Reducing the polynomial allows you to find the remaining roots quickly.

Always ensure your polynomial is written in standard descending order of exponents before extracting coefficients a₀ through a_n. If any intermediate coefficient is missing, remember to input a value of zero for that term. Double-check your signs when listing factors, as missing a negative sign can omit valid negative roots from your potential list.